SYSTEM, METHOD AND APPLICATION TO CONVERT TRANSDERMAL ALCOHOL CONCENTRATION TO BLOOD OR BREATH ALCOHOL CONCENTRATION

Abstract

System, method and application that obtains, consolidates, and integrates multiple sources of data including Transdermal Alcohol Concentration (TAC) along with drinking diary, photo/video, breath analyzer, other biological data (e.g., heart rate, skin conductance. blood flow, person-level biometrics), and environmental data (e.g., ambient temperature, humidity, GPS) and uses models described herein to convert TAC obtained from a wearable biosensor into estimated Blood Alcohol Concentration (BAC) or Breath Alcohol Concentration (BrAC).

Claims

1. A method for converting transdermal alcohol concentration (TAC) to blood or breath alcohol concentration (BAC/BrAC), the method comprising: measuring, using a biosensor, the TAC of a human; receiving, by a processor, data corresponding to one or more drinking curves for a population of humans; receiving, by the processor, data corresponding to at least one of (i) static characteristics of the human, (ii) physiological characteristics of the human, and (iii) current environmental conditions; and converting, using the processor, the TAC to BAC/BrAC using the data from one or more drinking curves, and the at least one of (i) the static characteristics of the human, (ii) the physiological characteristics of the human, and (iii) the current environmental conditions.

2. The method of claim 1, wherein the data corresponding to the one or more drinking curves includes a measurement of TAC and a measurement of at least one of BAC and BrAC.

3. The method of claim 1, wherein the data corresponding to the one or more drinking curves includes a time sequence of measurements of TAC and a time sequence of measurements of BAC or BrAC, and wherein the method is performed in real time.

4. The method of claim 1, wherein the data corresponding to the static characteristics includes a measurement of at least one of age, sex, ethnicity, height, weight, body fat and muscle, skin color, skin thickness, and skin tortuosity, wherein the data corresponding to the physiological characteristics includes a measurement of at least one of sweat, skin conductance, skin hydration, exercise, heart rate, blood pressure, blood flow, and stomach content, and wherein the data corresponding to the current environmental conditions includes a measurement of at least one of ambient temperature, humidity, pressure, GPS, weather, and climate.

5. The method of claim 1, wherein the converting is performed using a deterministic or stochastic finite dimensional autoregressive moving average with exogenous input (ARMAX) input/output model.

6. The method of claim 1, wherein the converting is performed using a blind or Bayesian deconvolution scheme.

7. The method of claim 1, wherein the converting is performed using a lattice filter-based recursive identification scheme.

8. The method of claim 1, wherein the converting is performed using an artificial neural network (ANN) by the processor, wherein the processor is remote from the biosensor and connected to the biosensor by a network.

9. The system of claim 1, wherein the converting is performed using a hidden Markov model (HMM) or a physics-informed hidden Markov model (PIHMM) by the processor.

10. The system of claim 1, wherein the converting is performed using a deconvolution filter based on output feedback linear quadratic Gaussian tracking gain computed by the processor.

11. The system of claim 1, wherein the converting is performed using first principles physics-based forward model with random parameters having distributions fit to population BrAC/TAC data and wherein the fitting the distributions is based on a na?ve pooled or mixed effects statistical model using either maximum likelihood, method of moments, or Bayesian techniques by the processor.

12. A system for converting transdermal alcohol concentration (TAC) to blood or breath alcohol concentration (BAC/BrAC), wherein the converting is in real-time with progressive forecasting and modeling techniques and recursive updating methods, the system comprising: a biosensor for measuring the TAC of a human; and a processor configured to: receive data from one or more drinking curves from a population of humans; receive data corresponding to at least one of (i) static characteristics of the human, (ii) physiological characteristics of the human, and (iii) the current environmental conditions; and convert, by the processor, in real-time the TAC to BAC/BrAC using the data from one or more drinking curves and the at least one of (i) the static characteristics of the human, (ii) the physiological characteristics of the human, and (iii) the current environmental conditions.

13. The system of claim 10, wherein the processor is remote from the biosensor and is connected to the biosensor via a network.

14. The system of claim 10, further comprising a remote database containing the one or more drinking curves from the population of humans connected to the processor via a network.

15. The system of claim 10, wherein the system comprises a plurality of further biosensors connected to the processor via a network, wherein the processor coverts, in real-time the TAC to BAC/BrAC for each of the plurality of further biosensors.

16. The system of claim 10, wherein the data corresponding to the one or more drinking curves includes a measurement of TAC and a measurement of at least one of BAC and BrAC.

17. The system of claim 10, wherein the data corresponding to the static characteristics includes a measurement of at least one of age, sex, ethnicity, height, weight, body fat and muscle, skin color, thickness, and tortuosity, wherein the data corresponding to the physiological characteristics includes a measurement of at least one of sweat, skin conductance, skin hydration, exercise, heart rate, blood pressure, blood flow, and stomach content, and wherein the data corresponding to the current environmental conditions includes a measurement of at least one of ambient temperature, humidity, pressure, GPS location data, weather, and climate.

18. The system of claim 10, wherein the converting is performed in real-time using a deterministic or stochastic finite dimensional autoregressive moving average with exogenous input (ARMAX) input/output model.

19. The system of claim 10, wherein the converting is performed using an artificial neural network (ANN) or a physics-informed neural network (PINN) by the processor.

20. A biosensor device for converting transdermal alcohol concentration (TAC) to blood or breath alcohol concentration (BAC/BrAC), the device comprising: a wearable sensor contactable to a human skin to measure the TAC of the human; a processor connected to the wearable sensor and connectable to a network, the processor configured to receive, via the network, data corresponding to one or more drinking curves for a population of humans; the processor configured to convert TAC to BAC/BrAC using (i) the data from one or more drinking curves and (ii) the measured TAC.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Other systems, methods, features, and advantages of the present invention will be or will become apparent to one of ordinary skill in the art upon examination of the following figures and detailed description.

[0013] FIG. 1A depicts a system for converting transdermal alcohol concentration (TAC) to blood or breath alcohol concentration (BAC/BrAC), in accordance with various embodiments;

[0014] FIG. 1B depicts a system for converting multiple TACs to BAC/BrAC for multiple biosensors, in accordance with various embodiments;

[0015] FIG. 1C depicts a method for converting transdermal alcohol concentration (TAC) to blood or breath alcohol concentration (BAC/BrAC), in accordance with various embodiments;

[0016] FIG. 2 depicts values of the {circumflex over (q)} estimators calculated from the simulated data for 20 observations, in accordance with various embodiments;

[0017] FIG. 3 depicts values of the {circumflex over (q)} estimators calculated from the simulated data for 60 observations, in accordance with various embodiments;

[0018] FIG. 4 depicts values of the {circumflex over (q)} estimators calculated from the simulated data for 100 observations, in accordance with various embodiments;

[0019] FIG. 5 depicts a range and distribution of BrAC observations, in accordance with various embodiments;

[0020] FIG. 6 depicts a range and distribution of TAC observations, in accordance with various embodiments;

[0021] FIG. 7 illustrates a chart 700 of BrAC, TAC observations and estimated BrAC that results from using the minimizer {circumflex over (q)}=(0.6341,0.7826), in accordance with various embodiments;

[0022] FIG. 8A shows the values for the loss functions over a number of iterations, in accordance with various embodiments;

[0023] FIG. 8B shows the of q.sub.1 and q.sub.2 over the number of iterations over the number of iterations, in accordance with various embodiments;

[0024] FIG. 9A shows a distribution for 88 selected drinking episodes using a standard normal distribution for the prior of the latent variable, in accordance with various embodiments;

[0025] FIG. 9B shows a distribution using the posterior over the latent variable, in accordance with various embodiments;

[0026] FIG. 9C displays the distribution for the full set of 126 drinking episodes with a standard normal prior for the latent variable, in accordance with various embodiments;

[0027] FIG. 9D shows the corresponding distribution using the posterior distribution for the latent variable, in accordance with various embodiments;

[0028] FIG. 10A shows a distribution of the posterior latent variable with a historgram for 88 drinking sessions used as training data, in accordance with various embodiments;

[0029] FIG. 10B shows the histogram for 126 drinking sessions used as training data, in accordance with various embodiments;

[0030] FIGS. 11A-D show results for four selected drinking episodes using the parameter distribution from training the GAN with all 126 drinking episodes, in accordance with various embodiments;

[0031] FIGS. 12A-F show a comparison of predicted BAC/BrAC signals using different drinking episodes, in accordance with various embodiments;

[0032] FIG. 13 depicts functional control gains, in accordance with various embodiments;

[0033] FIG. 14 depicts observer gains, in accordance with various embodiments; and

[0034] FIG. 15 depicts a chart with a shaded region is the 90% credible band centered at the mean for the optimal functional control gains ?.sub.3 computed using the disclosed method, in accordance with various embodiments.

DETAILED DESCRIPTION

[0035] A system, method and/or a mobile application (app) that converts Transdermal Alcohol Concentration (TAC) to estimated Blood or Breath Alcohol Concentration (BAC/BrAC) in real-time and post-drinking, by using a novel collection of data from biosensors, self-report, and the environment.

[0036] With the goal of well-founded statistical inference on an individual's blood alcohol level based on noisy measurements of their skin alcohol content, this disclosure develops M-estimation methodology in a general setting. Discussions herein then apply it to a diffusion equation-based model for the blood/skin alcohol relationship thereby establishing existence, consistency, and asymptotic normality of the nonlinear least squares estimator of the diffusion model's parameter and the resulting estimated blood alcohol curve. Simulation studies show agreement between the performance of these estimators and their asymptotic distributions, and the results are applied to a real skin alcohol data set collected via biosensor.

[0037] A goal is to model and estimate a human subject's alcohol concentration in the blood (BAC) or breath (BrAC) as a function of the alcohol level measured at the skin, i.e., the transdermal alcohol concentration (TAC), via a biosensor. Approximately 1% of the alcohol ingested in the human body is metabolized through the skin. For decades it has been recognized that the levels of TAC are connected to those of BAC/BrAC, but also that there are challenges in modeling this relationship. Because alcohol has to pass from the blood through the skin to be captured by a TAC sensor placed on the surface of the skin, it is subject to variation across individuals (e.g., skin layer thickness, porosity, tortuosity, etc.) and drinking episodes (e.g., ambient temperature, humidity, subject activity level, skin hydration, vasodilation, etc.). These effects result in a TAC-BAC/BrAC relationship that can be highly variable. Thus, TAC devices to date have typically been primarily used only in legal and research settings as abstinence monitors (e.g., in court mandated monitoring of DUI offenders) because of difficulties researchers have found translating raw TAC to the quantity of alcohol in the blood.

[0038] Still, TAC measured by a wearable biosensor device has great potential as a tool to improve personal and public health. It provides a passive, unobtrusive way to collect naturalistic data for extended periods of time. The same is not true about BrAC, which typically must be measured by trained research staff in the laboratory under controlled conditions using a breath analyzer, and thus is less practical for capturing alcohol levels in the field under real-world conditions. Moreover, the breath analyzer requires a user to be compliant, potentially interferes with naturalistic drinking patterns, and is subject to inaccuracy (e.g., readings too high due to mouth alcohol, or too low due to not properly taking a deep lung breath for a reading). Thus, creating a system that reliably converts TAC data into estimates of BAC (or BrAC) would greatly benefit the alcohol research and clinical communities who, along with public health institutes, have been quite interested in such models. Such a tool would dramatically improve the accuracy of field data and the validity of naturalistic studies of alcohol-related health outcomes, disease progression, treatment efficacy, and recovery. A wearable alcohol monitoring device could have consumer appeal as well, helping individuals monitor their own alcohol levels and make better health choices.

[0039] Work on the TAC-BAC/BrAC relationship begins with deterministic models for the forward process of the propagation of alcohol from the blood, through the skin, and its measurement by the sensor. Other approaches reverse the forward process to estimate BrAC based on the TAC. These efforts show unaccounted for variation in the TAC-BAC/BrAC relationship and subsequent work began to incorporate uncertainty into the models via a random diffusion equation. Other statistical modeling approaches include a regression model for peak BrAC using peak TAC, time of peak TAC, and gender using controlled laboratory data. Other efforts examine time delays from peak BrAC to peak TAC. Further efforts use physics-based statistical models for the TAC-BAC/BrAC relationship.

[0040] In this disclosure, systems, methods, and devices are presented to meet this challenge by using a physics-based statistical model that allows individual, device, and drinking episode level variation by treating the data from each person/device/episode triple as resulting from its own model parameters. Discussions herein determine the large sample behavior of estimates of these parameters and give conditions under which these estimates are consistent and have a limiting normal distribution. These discussions then use those results to give a statistically rigorous asymptotic characterization of the properties of the BrAC/BAC curve estimates obtained from measured TAC, including information on estimation error. As these estimates are made on an individualized basis, they will not be adversely affected when used in a study of a population whose characteristics vary widely. On the other hand, these estimatesin some embodimentsrequire individualized calibration over subject, device and environmental conditions

[0041] While the discussion includes calibration aspects, in further embodiments, the key model parameters depend on measurable subject and environmental covariates which may be measured, and which eliminates some or all calibration.

[0042] It may, in some embodiments be desirable to quantitatively estimate BAC/BrAC from TAC to within the desired degree of accuracy after first calibrating the underlying models to each individual subject and situation, thus accounting for confounding person-level, environmental, and physiological factors that differ across the population of subjects and across situations. The forward and inversion models included in the app are sophisticated mathematical systems that include deterministic and population models and supervised learning algorithms.

[0043] First, the forward model captures the dynamics of the transport of ethanol molecules from the blood through the skin and its measurement(s) by the biosensor. The app includes the option to calibrate the forward model based on individualized data obtained from a real-time drink diary, retrospective drink diary, or pre-set drinking paradigm, or based on population-based models alone or combined with individualized personal data (e.g., age, sex, ethnicity, skin, height, weight, body fat, etc.)

[0044] Then, in the inversion process the model is used to deconvolve estimated BAC/BrAC in future drinking episodes from measured TAC and all other available data. This means the BAC/BrAC in subsequent drinking episodes is estimated from the TAC provided by the biosensor without any further action by the user.

[0045] The real-time deconvolution scheme to estimate BAC/BrAC uses novel models that incorporate adaptive real-time data driven model refinement/learning, autoregressive moving average with exogenous input (ARMAX), and lattice filter-based recursive identification schemes to produce estimates in real-time, and which can be continuously updated with new data. An additional approach to recovering BAC/BrAC from TAC includes a real-time deconvolution scheme based on a technique from linear control and estimation theory. Farther mechanisms are also discussed.

[0046] Once drinking is complete for an episode and TAC has returned to or established a baseline, the app uses the full set of data to update the model BAC/BrAC estimates using the entire set of data for the episode. In addition, over time individuals can update their personalized model fits with data obtained through the app and paired biosensors in additional drinking sessions. As data accumulates for an individual subject, Bayesian techniques are used to improve the accuracy of the estimated BAC/BrAC. Finally, the app also includes components for capturing subjective responses to alcohol (e.g., feeling flushed, intoxicated) and drinking context (e.g., vis photos, video, GPS location) beyond alcohol consumption, using automated reminders, random prompts, and/or self-timed diary entries options, and this data can then be paired to estimated BAC/BrAC and other biosensor measurements.

[0047] The output includes TAC and estimated BAC/BrAC curves with credible bands, additional biosensor data and subjective ratings of alcohol response displayed alone and in conjunction with estimated BAC/BrAC, and summary scores of drinking events along with correlations with subjective ratings of alcohol response and drinking contexts. Summary scores will also be retained and displayed in a calendar format, which will also allow for retrospective recording of drinking sessions when not wearing a TAC biosensor. This multi-faceted app provides comprehensive assessment and result options for capturing drinking in real-time and consolidating this data into meaningful metrics. This multifaceted app provides a comprehensive system that incorporates all available data, utilizes self-report through a novel web application, and includes real-time forecasting of estimated BAC/BrAC curves and scores. This app is the first effective tool for non-experts to produce quantitative estimates of BAC/BrAC from TAC and other data.

[0048] A wearable biosensor (e.g., a digital watch, fuel cell, Fitbit?) may be used to measure or sense ethanol molecules from the blood via the skin. The system is based on a fit forward model in the form of a partial differential (diffusion) equation that captures the dynamics of the transport of ethanol molecules from the blood through the skin and its measurement by the biosensor. The system then uses the estimated model to deconvolve estimated BAC/BrAC from the biosensor measured TAC. The accuracy of the estimated BAC/BrAC is significantly improved by correcting for environmental and physiological factors that differ across the population of subjects and situations. Therefore, it is important that the underlying models be, in some form, calibrated to each subject, device, and situation The system utilizes sophisticated mathematical population models and supervised learning algorithms together with the capability to optionally enter drinking diary, breathalyzer, and other biosensor data to tune the underlying models to the physiological characteristics of the person wearing the device and the current environmental conditions. The BAC/BrAC for all drinking episodes can then be estimated from the TAC passively provided by the biosensor without any active participation by the user.

[0049] This invention extends the scope of application of TAC to BAC/BrAC conversion software and adjusts for variations (i) between subjects, (ii) within subjects, (iii) in environmental conditions, (iv) across hardware devices, and/or (v) in repeated measurements over time, when applying the diffusion model, and (vi) can be fit in real-time. In particular, the invention utilizes statistical models for the low dimensional input parameters to the diffusion equations that depend on covariate information that describe characteristics of the subjects and their environment. The end result is personalized, real-time BAC/BrAC estimates with accompanying statistical accuracy measures, such as credible intervals and margins of error. In addition, the invention provides a theoretical, asymptotic analysis of the performance of the new estimation methods that result upon embedding the models in the underlying diffusion equation.

[0050] The invention utilizes adaptive real-time data driven model refinement/learning For example, the invention has the ability to incorporate real-time drink diary data into one or more of the underlying physics-based models described earlier to construct an adaptive/recursive data assimilation, estimation, and prediction system. The models are continuously updated with newly available real-time individual-level data to produce revised/estimated BAC/BrAC based on TAC in real-time. Even though the underlying state equation that forms the basis of this invention is, in general, infinite dimensional, end-to-end, it is a single input/single output linear time invariant system. Thus, BAC/BrAC can be approximated using a deterministic or stochastic finite dimensional autoregressive moving average with exogenous input (ARMAX) input/output model. The invention further includes lattice filter-based recursive identification schemes, which allow for the efficient modification of both the order of the model and the parameters when new data is introduced into the system. The invention takes advantage of the wealth of real-time adaptive parameter estimation, filtering, prediction, and deconvolution schemes available for systems described by these types of models. The invention accounts for the introduction of nonlinearities into these schemes through the use of artificial neural networks (ANNs) and trains them using a variant of back propagation. This scheme yields a somewhat delayed estimated BAC/BrAC, which can then be augmented by a prediction scheme to yield preliminary real-time estimated BAC/BrAC, and afterwards update estimated BAC/BrAC for the entire episode.

[0051] The invention incorporates new innovations that serve to improve the efficiency and accuracy of the estimated BAC/BrAC. In particular, two approaches to deconvolving the BAC/BrAC signal from the TAC signal have been included. One approach used to recover BAC/BrAC from TAC is based directly on our physiological model for the diffusion of ethanol through the epidermal layer of the skin. While this approach provides an effective low rank parameterization of the relationship between BAC/BrAC and TAC when there was extremely limited experimental data, a more empirical model can offer more flexibility when a relatively rich pool of laboratory collected contemporaneous matched BAC/BrAC-TAC data is available.

[0052] In an empirical linear model, we assume that the measured TAC is the convolution of two random signals, the convolution kernel K(s; ?) and the measurement error ?(s; ?). That is,

y.sub.TAC(t; ?)=?.sub.??.sup.tK(t?s; ?)v.sub.BrAC(s)ds+?(s; ?).

[0053] The process of training the model given in the above equation consists of identify reliable distributions for the functions K and ? based on available matched BAC/BrAC-TAC pairs. Since both functions belong to an infinite-dimensional space of random functions, effective parameterization of these function spaces is crucial to ensure stability of the training process. Inspired by the physiological model, a family of cubic spline functions defined on a strategically selected non-uniform grid is chosen. Analysis of the optimally determined kernel functions from a set of BAC/BrAC-TAC pairs exhibited an encouraging level of consistency among test subjects and data from different sessions for the same test subject.

[0054] Using the resulting distributions for K and ? obtained through analysis of data for an appropriate cohort or population, the retrieval of BAC/BrAC from TAC is done in bear real-time by calculating statistically consistent and efficient estimators for BAC/BrAC. One example of such an estimator is given by

[00001]${\stackrel{\_}{v}}_{\mathrm{BrAC}}=\arg \underset{v_{\mathrm{BrAC}}}{\min }\left(\underset{k=1}{\overset{n}{.Math.}}{.Math.{\stackrel{\_}{y}}_{\mathrm{TAC}}\left(t_k\right)-{?}_{-?}^{t_k}K\left(t_k-s;?\right)v_{\mathrm{BrAC}}\left(s\right)\mathrm{ds}.Math.}^2+{.Math.K\left(.Math.;?\right)-\stackrel{\_}{K}(.Math.).Math.}^2\right),$

where y.sub.TAC(t.sub.k) represents the measured TAC value at time t.sub.k and K corresponds to the population mean for the kernel functions. Note that in the optimization above, the calculation obtains an optimal pair of estimators. v.sub.BrAC and K(?; ?) As data accumulates for an individual subject, Bayesian techniques are used to improve the accuracy of the retrieved BAC/BrAC signal.

[0055] Another approach to recovering BAC/BrAC from TAC includes a real-time deconvolution scheme based on a linear control and estimation theory technique. By formulating the deconvolution problem as a linear quadratic Gaussian tracking problem, the estimated BAC/BrAC signal is obtained in the form of a linear output feedback law. More precisely, the estimated BAC/BrAC signal is given as a real-time linear function of the measured TAC signal. Undesirable non-physical oscillations in the estimates which result from the underlying ill-posedness of the filtering problem being solved to determine the BAC/BrAC signal are mitigated by including an appropriate penalty term in the quadratic performance index. This approach also yields credible bands and error bars along with the estimated BAC/BrAC signal.

[0056] Beyond the mathematical models, this software invention includes real-time and retrospective self-report data collection mobile app for recording drinking diary, breathalyzer, other biosensor data, drinking context, and other factors that vary over a drinking episode (e.g., stomach contents, mood, behavior). The app includes the option to add calibration data from individualized data obtained from a real-time drink diary, retrospective drink diary, pre-set drinking paradigm, or based on population-based models combined with individualized personal data (e.g., age, sex, ethnicity, skin, height, weight, body fat, etc.). The app also includes components for capturing subjective responses to alcohol (e.g., feeling flushed, intoxicated) and drinking context (e.g., via photos, video, GPS location) beyond alcohol consumption, using automated reminders, random prompts, and/or self-timed diary entries options, and these data can then be paired to estimated BAC/BrAC and other biosensor measurements. Summary scores of drinking events along with correlations with subjective ratings of alcohol response and drinking contexts will be calculated and displayed in episode-level figures and charts These summary scores will also be retained and displayed for multiple drinking episodes in a calendar format, which also will allow for retrospective recording of drinking sessions.

[0057] The invention is implemented using a combination of hardware and software. The hardware includes the TAC biosensor, processors, memories, displays, and environmental sensors. The software includes computer code that can run on the hardware. The invention allows the user the option to select which method(s) they would like to use through a set of menus, based on what the user prioritizes to optimize, similar to factor analyses options in commercially available statistical software where the user can select different matrix rotations or fit indices to emphasize in the model runs. The invention produces both estimated BAC/BrAC curves, credible bands, and summary scores such as maximum estimated BAC/BrAC, time of maximum BAC/BrAC and area under the BAC/BrAC curve.

[0058] With reference to FIG. 1A, system 2 for converting transdermal alcohol concentration (TAC) to blood or breath alcohol concentration (BAC/BrAC) in real-time may include a biosensor 6 connected to a backend control system 4. In various embodiments, the biosensor is a wearable device. In further instances, the biosensor is a combination of devices interconnected by a body area network. For instance, aspects of the biosensor may be worn adjacent a user's skin and other aspects of the biosensor may be carried in a pocket, a purse, or otherwise near or about the person of a user. The biosensor 6 measures a biological indicator, such as ethanol present in sweat or on/in a user's skin. The biosensor 6 may provide data representative of the biological indicator to a backend control system 4 for processing and may receive in return, an indication of the user's BAC/BrAC. In further instances, the biosensor 6 is not connected to a backend control system 4 and instead performs calculations on a local processor to determine an indication of the user's BAC/BrAC. In various embodiments, the biosensor 6 is selectably connectable to the backend control system 4. For instance, the biosensor 6 may perform calculations locally when disconnected from the backend control system 4 or may provide data to the backend control system 4 for the performance of calculations by the backend control system when connected to the back end control system 4. In further embodiments, the biosensor 6 stores data representative of the biological indicator when disconnected from the backend control system 4 and provides this data to the backend control system 4 when a connection is established. In this manner, the biosensor 6 may measure a TAC and the biosensor 6 and/or a backend control system 4 may calculate a corresponding BAC/BrAC. In various embodiments, the biosensor 6 and/or the backend control system 4 may display the corresponding BAC/BrAC in human readable form, such as on a display terminal.

[0059] The biosensor 6 may include a sensor 10. The sensor 10 may include an element configured to measure a TAC. For instance, a fuel cell device may process ethanol present on a user's skin or in a user's sweat to generate electricity, which may be measured. Because the voltage, current, power, and/or other measurable aspect of the generated electricity may be quantified, the corresponding amount of ethanol responsible for generating the electricity may be quantified. Various references to sensor 10 elsewhere herein provide example sensors for various embodiments.

[0060] The biosensor 6 may include a processor 20. The processor may be a computer, of a microcontroller, or a low power embedded microprocessor, or a single-board computer, an application-specific integrated circuit (ASIC) or any other electronic data processing device as desired. In various embodiments, the processor 20 is connected to a memory 80. The memory 80 may be a working memory, providing for data storage during calculation by the processor 20 of BAC/BrAC from TAC. The memory 80 may be a storage memory, such as for storage of data corresponding to TAC prior to transmission of this data to a backend control system 4. The memory maybe both a storage memory and a working memory.

[0061] The biosensor 6 may have a local display terminal connected to the processor 20. The local display terminal 30 may be a human-readable interface. For instance, the local display terminal 30 may be one or more LED, audio annunciator, tactile feedback device, LCD or other text or graphic display, or any other apparatus configured to provide information in human-readable form. In various embodiments, the local display terminal provides menu structures and other interface elements of an application as described herein. In various embodiments, the local display terminal displays a TAC measurement. In further embodiments, the local display terminal displays a calculated BAC/BrAC measurement calculated by the biosensor 6, the backend control system 4, or a combination of the biosensor 6 and the backend control system 4 that is calculated from a measured TAC.

[0062] The biosensor 6 may be connectable to a network 70. The backend control system 4 may also be connectable to the network 70. The network 70 may permit electronic communication between the biosensor 6 and the network 70. In various embodiments, the network 70 comprises the internet. In further embodiments, the network 70 may be a private network, or a virtual private network, or an RF data link, or an optical data link, or a wired link, or any electronic connection. The network 70 may include wireless aspects, such as cellular connections, or Wi-Fi connections or other aspects.

[0063] Having discussed the biosensor 6 and a network 70, attention is now directed to a backend control system 4. The backend control system 4 may comprise a server, or a cloud computing resource, or any other computing system as desired. In various instances, the backend control system 4 provides greater processing power than the biosensor 6 and facilitates calculation of BAC/BrAC from TAC by remotely handling calculations and other processing tasks. In various instances, the backend control system 4 collects and aggregates data from the biosensor 6 with data from other resources, such as user inputs, stored or laboratory research data, previously collected data such as prior TAC data, user-specific data such as weight, height, and other aspects, training data, and/or the like. In various instances, the backend control system 4 collects and aggregates data from multiple different biosensors 6. Various data, factors, and relevant variables are discussed throughout, each of which may be processed. stored, or otherwise received by the backend control system 4 and/or the biosensor 6.

[0064] The backend control system 4 may include a remote database 50. The remote database 50 may store the aforementioned data, TAC calculations, BAC/BrAC calculations and/or the like. The remote database 50 may provide both working memory and/or storage memory.

[0065] The backend control system may include a remote processor connected to the remote database 50 and the network 70. The remote processor may a computer, or a microcontroller, or a low power embedded microprocessor, or a single-board computer, an application-specific integrated circuit (ASIC) or any other electronic data processing device as desired. The remote processor may be a distributed or cloud computing resource.

[0066] The backend control system 4 may have a remote display terminal 40 connected to the remote processor 60. The remote display terminal 40 may be a human-readable interface. For instance, the remote display terminal 40 may be one or more LED, audio annunciator. tactile feedback device, LCD or other text or graphic display, or any other apparatus configured to provide information in human-readable form. In various embodiments, the remote display terminal 40 provides menu structures and other interface elements of an application as described herein. In various embodiments, the remote display terminal 40 displays a TAC measurement. In further embodiments, the remote display terminal 40 displays a calculated BAC/BrAC measurement calculated by the biosensor 6, the backend control system 4, or a combination of the biosensor 6 and the backend control system 4 that is calculated from a measured TAC. The remote display terminal 40 may be separate from the backend control system 4 and connected to the network 70. The remote display terminal 40 may be browser session of a user accessing the backend control system 4, such as via a website login interface on an internet browser running on a commodity personal computer.

[0067] Previously, it was mentioned that the backend control system 4 may collect and aggregate data from multiple different biosensors 6. In addition, the backend control system 4 may provide processing resources to multiple different biosensors for calculating a BAC/BrAC from a measured TAC. With reference to FIG. 1B, in various instances, a backend control system 4 is connected to a first biosensor 6-1, a second biosensor 6-2, and a third biosensor 6-3. The backend control system 4 may be connected to any number of biosensors. While not illustrated in FIG. 1B, in various embodiments, the biosensors and the backend control system 4 may be connected via a network 70.

[0068] Thus, in various instances, the system 2 for converting transdermal alcohol concentration (TAC) to blood or breath alcohol concentration (BAC/BrAC) in real-time may include a biosensor 6 for measuring the TAC of a human. The system may include a processor. The processor may be a processor 20, a remote processor 60, or a combination of the processor 20 and remote processor 60 such that certain processes are conducted on processor 20 and other processes are conducted on remote processor 60. As such, one or more of the processors may receive data from one or more drinking curves from a population of humans. One or more of the processors may receive data corresponding to at least one of (i) static characteristics of the human, (ii) physiological characteristics of the human, and (iii) the current environmental conditions. One or more of the processors may convert in real-time the TAC to BAC/BrAC using the data from one or more drinking curves and the at least one of (i) the static characteristics of the human, (ii) the physiological characteristics of the human, and (iii) the current environmental conditions.

[0069] Moreover, the biosensor 6 for converting transdermal alcohol concentration (TAC) to blood or breath alcohol concentration (BAC/BrAC) may include a wearable sensor 10 contactable to a human skin to measure the TAC of the human and a processor (processor 20, remote processor 60, and/or a combination of processor 20 and processor 60) connected to the wearable sensor 10 and connectable to a network 70, the processor configured to receive, via the network 70, data corresponding to one or more drinking curves for a population of humans. One or more of the processor may be configured to convert TAC to BAC/BrAC using (i) the data from one or more drinking curves and (ii) the measured TAC.

[0070] Turning now to FIG. 1C, a method 100 of calculating a BAC/BrAC from TAC may be provided. One may appreciate that the various calculations discussed elsewhere herein may be implemented by such a method 100 and such a method 100 may be implemented by the embodiments of FIG. 1A-B. A method 100 for converting transdermal alcohol concentration (TAC) to blood or breath alcohol concentration (BAC/BrAC) may include multiple steps. For instance, the method may include measuring, using a biosensor, the TAC of a human (block 102). The method may include receiving by a processor data corresponding to one or more drinking curves for a population of humans (block 104). Notably, the processor may be a processor local to the biosensor 6 (FIG. 1A, processor 20) or may be a remote processor (FIG. 1A, remote processor 60).

[0071] The method may include receiving, by a processor, data corresponding to at least one of (i) static characteristics of the human, (ii) physiological characteristics of the human, and (iii) current environmental conditions (block 104). Again, the processor may be a processor local to the biosensor 6 (FIG. 1A, processor 20) or may be a remote processor (FIG. 1A, remote processor 60). This processor may be a different processor than that referred to in block 104.

[0072] Finally the method may include converting, using a processor, the TAC to BAC/BrAC using the data from one or more drinking curves, and the at least one of (i) the static characteristics of the human, (ii) the physiological characteristics of the human, and (iii) the current environmental conditions (block 106). This processor may be the processor 20 (FIG. 1A) or remote processor 60 (FIG. 1B) and may be a same or different processor as that of blocks 104 and/or 106.

[0073] Various methods for such converting are discussed at length throughout this disclosure. For instance, the converting may be performed using a deterministic or stochastic finite dimensional autoregressive moving average with exogenous input (ARMAX) input/output model. The converting may be performed using a blind or Bayesian deconvolution scheme. The converting may be performed using a lattice filter-based recursive identification scheme. The converting may be performed using an artificial neural network (ANN). The converting may be performed using a hidden Markov model (HMM) or a physics-informed bidden Markov model (PIHMM). The converting may be performed using a deconvolution filter based on output feedback linear quadratic gaussian tracking gain. Moreover, the converting may be performed using first principle physics-based forward models with random parameters having distributions fit to population BrAC/TAC data. The fitting the distributions may be based on a naive pooled or mixed effects statistical model using either maximum likelihood, method of moments, or Bayesian techniques. The converting may be performed in many different ways. The converting may be performed in real-time with progressive forecasting and modeling techniques and recursive updating.

[0074] Thus, one may appreciate that the method may have various additional aspects. For instance, the data corresponding to the one or more drinking curves may be different types of data. The data may be a measurement of TAC. The data may be a measurement of BAC. The data may be a measurement of BrAC. The data may include comparisons of TAC to BAC and/or BrAC.

[0075] The data that corresponds to the static characteristics may include a variety of different measurements. For instance, the measurements may relate to aspects of a specific human for whom TAC is being measured. The measurements may include a measurement of at least one of age, sex, ethnicity, height, weight, body fat and muscle, skin color, skin thickness, and skin tortuosity.

[0076] The data that corresponds to the one or more physiological characteristics may include a variety of different measurements. For example, the measurements may relate to aspects of a specific human for whom TAC is being measured but which may be dynamic. For instance, the measurements may include a measurement of at least one of sweat, skin conductance, skin hydration, exercise, heart rate, blood pressure, blood flow, and stomach content.

[0077] The data that corresponds to the current environmental conditions may include a variety of different measurements. For example, the measurements may relate to aspects of an environment that the human for whom TAC is being measured is exposed to. The measurements may include a measurement of at least one of ambient temperature, humidity, pressure, GPS, weather, and climate.

[0078] Having provided an overview of the system, method, and device above, attention is now directed to a discussion of a diffusion model to characterize ethanol diffusion across skin so that correspondingly the subject's BAC/BrAC may be model as a function of TAC.

[0079] The following discussion will include various types of models. For instance, a partial differential equation diffusion model may characterize alcohol transfusion across the skin. A least squares approach may be provided for estimating an unknown vector. M-estimation is provided and basic examples of its use, as well as an application of M-estimation to the mentioned model. Yet further, the application of the M-estimation to the partial differential equation diffusion model may be implemented to obtain results on the performance of resulting BrAC curves estimated from TAC. The discussion will also include an evaluation of theoretical results in simulations and an illustration using BrAC/TAC relationships measured experimentally.

[0080] Diffusion Model (Section 1). Although a goal is to model a human subject's BAC/BrAC as a function of TAC, the ethanol molecules themselves move in the other direction: from the blood, through the skin, to ultimately be measured by the sensor on the surface of the skin. Thus the relevant physics describe the TAC as a function of BAC/BrAC. Consider a specific model for this transport based on Fick's law of diffusion which depends on an unknown, 2-dimensional parameter q=(q.sub.1, q.sub.2). The result is TAC expressed as a convolution of BAC/BrAC with a kernel or filter, and as a function of the unknown q which may then be estimated via nonlinear least squares as described and whose properties are considered in Section 3. These properties determine the inferential consequences for BAC/BrAC estimation, and in particular have a large impact on the accuracy of the estimated BrAC curve, as studied in Section 3.

[0081] Let x(t, ?) denote the concentration of ethanol at time t?0 and depth ?? [0,1] from the skin surface through epidermis, choosing units so that ?(t)=x(t, 1), t?0 is the BAC at time t. A Fick's law-based model has been developed and used successfully to model data of this type. The model specifies x(t, ?) as the solution to the partial differential equation, with boundary condition

[00002]$\begin{array}{cc}\frac{?x}{?t}=q_1\frac{{?}^{2}x}{?{?}^2},q_1\frac{?x}{??}{|}_{?=1}=q_2?\left(t\right),q_1\frac{?x}{??}{|}_{?=0}=x{|}_{?=0},& \left(1\right)\end{array}$

depending on the parameter q=(q.sub.1, q.sub.2). The TAC at skin level is then x(t, 0). When we want to emphasize dependency on the parameter q we will write, for instance, ?(t; q).

[0082] The system with its boundary conditions can be solved in continuous time in terms of unbounded linear operators, with solution

x(t)=e.sup.A(q)tx(0)+?.sub.0.sup.te.sup.A(q)(t?s)B(q)?(s)ds.(2)

[0083] In cases we consider, x(0) will be the zero function, that is, observation begins at, or before, the time of first intake of alcohol. By taking a discretization of the distance ? from skin level into k steps for some k sufficiently large, the operators in (2) can be approximated by k dimensional linear operators (i.e., matrices) yielding the approximation to the solution given by

x.sup.(k)(t)=?.sub.0.sup.te.sup.A(k).sup.(q)(t?s)B.sup.(k)(q)?(s)ds.(3)

[0084] Now fixing, and suppressing in the notation, the level of discretization k, an observation taken at time t can be represented as the linear function of x(t) given by

ds,(4)

plus an additive error term. For observations taken at skin level, the vector C will have a one in its first component, and zeros elsewhere.

[0085] The matrices in (4) depend on the unknown parameter q as

A(q)=q.sub.1D+E and B(q)=q.sub.2F,(5)

where C, D, E, and F are known matrices that result from making the finite-dimensional approximation discussed. More precise assumptions and properties of these matrices, and the domain of q, will be specified in Section 3.

[0086] Non-Linear Least Squares Estimation (Section 1.2). To estimate the parameter q, we assume that TAC data {y.sub.ij, 1?i?n, 1?j?m.sub.i} is collected on a single individual over n different drinking episodes at the my times 0?t.sub.i,1< . . . <t.sub.i,m.sub.i?T.sub.i, for given BrAC curves ?.sub.i on [0, T.sub.i]. With m=(m.sub.1, . . . , m.sub.n), the estimator minimizes

[00003]$\begin{array}{cc}{J}_{n,m}\left(q\right)=\frac{1}{2{.Math.}_{i=1}^n{m}_i}{.Math.}_{i=1}^n{.Math.}_{j=1}^{m_i}{\left(f_{{?}_i}\left(t_{\mathrm{ij}};q\right)-y_{\mathrm{ij}}\right)}^2,& \left(6\right)\end{array}$

where ?.sub.?.sub.i(t.sub.ij; q) is given by the right hand side of (4) with ? replaced by ?.sub.i, the BrAC curve for drinking episode i. The model specified by (4) and (5) is deterministic, but to account for measurement variability, we include additive, homoscedastic errors on the observed values TAC values. The constant variance condition implies that all TAC observations are equally reliable, and that the error variances, in particular, do not depend on the length of time elapsed since the last observation. For that reason, the least squares objective functions give equal weight to their summands, and when appropriate, weights, inversely proportional to variance, could be included. We may also allow the length of the time interval T.sub.i of the i.sup.th episode, and the location of the sampling times, to be stochastic.

[0087] In Section 2 below, we consider the existence, consistency, and limiting distribution of our least squares estimators in a general M-estimation context, and present some examples. In Section 3 we apply the results in Section 2 to the diffusion model of Section 1, and present Theorems 3.1 and 3.3, which contain our main results on inference for the main parameter q of interest, and also for the error variance ?.sup.2. In Section 4 we apply the results of Section 3 for making inference on the BrAC curve, and in particular for the construction of uniform error bounds on the resulting curve estimate. We validate our theoretical work via simulation and real data analysis in Section 5.

[0088] M-Estimation (Section 2)Existence, Consistency, and Limiting Distribution. In this section we consider M-estimation in a general setting that contains what we will require to handle the diffusion model we consider. Prior discussions of M-estimation tend to focus on the case of a univariate parameter, whereas ours covers the multivariate case. Prior efforts cover only least squares estimation whereas our results apply to the more general estimating equation (7). Also, previous results only apply to approximate normality and require i.i.d. error terms, whereas our Theorem 2.2 can be applied to other limiting distributions and relaxed conditions on the error terms, although our main application is to limiting normality. Finally, previous results are more restrictive in terms of a number of technical conditions, such as compactness of the parameter space ? which our results do not require, and the existence of tail products of vectors of observation means and error terms, which our results eschew in favor of more conventional regularity conditions on the score type function U.sub.n.

[0089] After establishing the notation and setup in Section 2.1, we state our main results in Section 2.2. In Section 2.3 we provide some general examples of the applications of our results to least squares and maximum likelihood estimation.

[0090] Set Up and Summary of Results (Section 2.1). For n?1, observed data X.sup.(n) in a space ?.sup.(n), a parameter space ??.sup.p having non-empty interior, and a function U.sub.n:???.sup.(n).fwdarw..sup.p, consider the estimating equation

U.sub.n(?)=0, ???,(7)

where the dependence of .sub.n on the data is suppressed. In our examples ?.sup.(n) will a Euclidean space endowed with a family of densities p.sub.n(x.sup.(n); ?), ??? which generate the data from this family with ?=?.sub.0. Two important situations in which the solutions of such equations arise are maximum likelihood and least squares estimation.

[0091] For maximum likelihood, under smoothness conditions on the densities, the maximizer of the log likelihood L.sub.n(?)=log p.sub.n(x.sup.(n); ?) is given as a solution to (7) with

.sub.n(?)=?.sub.?L.sub.n(?; X.sup.(n)),(8)

where ?.sub.? denotes taking derivative with respect to ?, resulting in a column vector of partial derivatives when ? itself is a vector. When the data X.sup.(n) consists of n independent random vectors X.sub.1, . . . , X.sub.n in .sup.d, each with distribution p(x; ?.sub.0), the space ?.sup.(n) can be identified with .sup.d?n, and p.sub.n(x.sup.(n), ?) is the product of the marginal densities p(x.sub.i?) for i=1, . . . , n.

[0092] To introduce least squares estimation, suppose that pairs (x.sub.i, y.sub.i)?.sup.d?, i=1, . . . , n, are observed with distribution depending on ? for which .sub.?[y.sub.i|x.sub.i]=?.sub.i(x.sub.i; ?) for ?.sub.i(x; ?) in some parametric class of functions. With x.sup.(n)=(x.sub.1, . . . , x.sub.n), the least squares estimate of ? is given as the minimizer of

[00004]$J\left(?;x^{\left(n\right)}\right)=\frac{1}{2n}{.Math.}_{i=1}^n{\left(y_i-f_i\left(x_i;?\right)\right)}^2,$

which under smoothness conditions can be obtained via (7) with

[00005]$\begin{array}{cc}{?}_n\left(?\right)={?}_{?}J\left(?;x^{\left(n\right)}\right)=\frac{1}{n}{.Math.}_{i=1}^n\left(f_i\left(x_i;?\right)-y_i\right){?}_{?}{f}_i\left(x_i;?\right),& \left(9\right)\end{array}$

[0093] The aim of the estimating equation U.sub.n(?)=0 is to provide a value close to the one where the function U.sub.n(?) takes the value of 0 in some expected, or asymptotic, sense. In particular, in Theorem 2.1 we will show, under that when U.sub.n(?.sub.0) is, under an appropriate scaling, close to zero as n.fwdarw.?, then the sequence of estimates obtained via the estimating equations will be consistent for the true parameter.

[0094] In Theorem 2.2, we will also provide a corresponding limiting distribution for solutions to the estimating equation (7). Let U.sub.n(, ?) have components

U.sub.n(?)=(U.sub.n,j(?)).sub.1?j?p where U.sub.n,j:.sup.n??.fwdarw..

[0095] In the case of maximum likelihood estimation, where we have (8), under the assumption of the existence and continuity of second derivatives of L.sub.n for ???, writing U.sub.n/(?) as short for the observed information matrix ?.sub.?U.sub.n.sup.T(?)?.sup.p?p, its k, j.sup.th component is given by

[00006]$\frac{?U_{n,j}\left(?\right)}{?{?}_k}=\frac{{?}^2{L}_n\left(?\right)}{?{?}_k?{?}_j}=\frac{{?}^2{L}_n\left(?\right)}{?{?}_j?{?}_k}=\frac{?U_{n,k}\left(?\right)}{?{?}_j}.$

[0096] And in this case, the third condition in (11) below is equivalent to the condition that the limiting information matrix l is positive definite. Tolerating a slight abuse of notation, we may also write ?.sub.j rather than ?.sub.?.sub.j when taking a partial with respect to the j.sup.th coordinate variable, and ?.sub.j.sup.m for the m.sup.th order derivative, for instance, denoting the k, j.sup.th entry of U.sub.n(?) by ?.sub.kU.sub.n,j(?).

[0097] Over each coordinate j=1, . . . , p, under second order differentiabilty conditions, we will make use of the second order Taylor expansion of U.sub.n,j(?) around some ?.sub.0??,

[00007]$\begin{array}{cc}{U}_{n,j}\left(?\right)=U_{n,j}\left({?}_0\right)+\underset{k=1}{\overset{p}{.Math.}}{?}_k{U}_{n,j}\left({?}_0\right)\left({?}_k-{?}_{k,0}\right)+\frac{1}{2}{.Math.}_{1?k,l?p}\left({?}_k-{?}_{k,0}\right){?}_{k,l}{U}_{n,j}\left({?}_{n,j}^{*}\right)\left({?}_l-{?}_{l,0}\right),& \left(10\right)\end{array}$

where each 0*.sub.n,j lies on the line segment connecting ? and ?.sub.0. In the following, we let ??? denote the Euclidean norm of a vector, the operator norm of a matrix, and the supremum norm of a function.

[0098] Estimating equations, consistency, and asymptotic normality (Section 2.2). We now present results that provide conditions for the consistency and existence of a non-trivial limiting distribution for a properly centered and scaled sequence of estimating equation solutions. We also include results on the consistent estimation of parameters on which the asymptotic distribution of our estimate may depend.

[0099] Theorem 2.1 Suppose that U.sub.n:???.sup.(n).fwdarw..sup.p is twice continuously differentiable in an open set ?.sub.0?? containing ?.sub.0, and that there exist a sequence of real members a.sub.n, a matrix ??.sup.p?p and ?>0 such that

[00008]$\begin{array}{cc}{a}_n{U}_n\left({?}_0\right){.fwdarw.}_{p}0\mathrm{and}{a}_n{U}_{n^{}}\left({?}_0\right){.fwdarw.}_p?\mathrm{as}& \left(11\right)\end{array}$$n.fwdarw.?\mathrm{with}\underset{.Math.?.Math.=1}{\inf }{?}^T\mathrm{??}=?.$

[0100] Suppose further that for any ??(0,1), that there exists a K such that for all n sufficiently large,

P(|a.sub.n?.sub.k,lU.sub.n,j(?)|?K, 1?k,l,j?p, ???.sub.0)?1??.(12)

[0101] Then for any given ?>0 and ??(0,1), for all n sufficiently large, with probability at least 1?? there exists {circumflex over (?)}.sub.n?? satisfying U.sub.n({circumflex over (?)}.sub.n=0 and ?{circumflex over (?)}.sub.n??.sub.0???, that is, a sequence of roots to the estimating equation (7) consistent for ?.sub.0.

[0102] In addition, for any sequence {circumflex over (?)}.sub.n.fwdarw..sub.p?.sub.0, we have

a.sub.nU.sub.n({circumflex over (?)}.sub.n).fwdarw..sub.p?,(13)

that is, ? can be consistently estimated by a.sub.nU.sub.n({circumflex over (?)}.sub.n) from any sequence consistent for ?.sub.0.

[0103] Proof: By replacing U.sub.n by a.sub.nU.sub.n and ? by ???.sub.0, we may assume that the conditions of Theorem 2.1 hold with a.sub.n=1 and ?.sub.0=0. For ?>0 let

B.sub.?={?:?????}.

[0104] For the given ??(0,1), let K and n.sub.0 be such that (12) holds with ? replaced by ?/2 for n?n.sub.0. For the given ?>0, take ??(0, ?) such that B.sub.???.sub.0 and C?<? where

[00009]$C=2+\frac{{\mathrm{Kp}}^{3/2}}{2}.$

[0105] By (11) there exists n.sub.1?n.sub.0 such that for n?n.sub.1

P(?U.sub.n(0)?<?.sup.2)?1??/3 P(?U.sub.n(0)??<?)?1??3,(14)

and also taken large enough so that (12) holds with ? replaced by ?/3. By the union bound, all three events hold with probability at least 1??. For ??B.sub.? and ?*.sub.n,j given by (10), the components of R.sub.n(?)=(R.sub.n,1(?), . . . , R.sub.n,p(?)).sup.T as defined by

[00010]$R_{n,j}\left(?\right)=\underset{1?k,l?p}{.Math.}{?}_k{?}_{k,l}{U}_{n,j}\left({?}_{n,j}^{*}\right){?}_l\mathrm{satisfy}.Math.R_{n,j}\left(?\right).Math.?{K\left(\underset{i=1}{\overset{p}{.Math.}}.Math.{?}_i.Math.\right)}^2?\mathrm{Kp}{.Math.?.Math.}^2.$

[0106] Then, for n?n.sub.1, with probability at least 1??, from (10), (14) and (12),

[00011]$\begin{array}{cc}.Math.U_n\left(?\right)-?\left(?\right).Math.?.Math.U_n\left(?\right)-U_{n^{}}\left(0\right)?.Math.+.Math.U_{n^{}}\left(0\right)?-\mathrm{??}.Math.=.Math.U_n\left(0\right)+\frac{1}{2}{R}_n\left(?\right).Math.+.Math.\left(U_{n^{}}\left(0\right)-?\right)?.Math.<{?}^2+\frac{{\mathrm{Kp}}^{3/2}}{2}{.Math.?.Math.}^2+?.Math.?.Math.?C{?}^2.& \left(15\right)\end{array}$$\mathrm{So}.Math.{?}^T{U}_n\left(?\right)-{?}^T\mathrm{??}.Math.<C{?}^3.$$\mathrm{Hence},\mathrm{if}.Math.?.Math.=?,{?}^T{U}_n\left(?\right)>{?}^T\mathrm{??}-C{?}^3?{\mathrm{??}}^2-C{?}^3={?}^2\left(?-C?\right)>0.$

[0107] Assume for the sake of contradiction that U.sub.n(?) does not have a root in B.sub.?. Then for ??B.sub.?, the function ?(?)=??U.sub.n(?)/?U.sub.n(?)? continuously maps B.sub.? to itself. By the Brouwer fixed point theorem, there exists ??B.sub.?, with ?(?)=?. Since ??(?)?=? for all ??B.sub.?, we have ??(?)?=???=?, contradicting (15) via ?.sup.2=???.sup.2=?.sup.T?=?.sup.T?(?)<0. Hence U.sub.n(?) has a root within ? of 0, and since ?<?, therefore within ?, with probability at least 1??, as required.

[0108] To prove (13), taking {circumflex over (?)}.sub.n to be any consistent sequence for ?.sub.0, a first order Talyor expansion yields, for all 1?j, k?p,

[00012]$\begin{array}{c}{?}_k{U}_{n,j}\left({\widehat{?}}_n\right)={?}_k{U}_{n,j}\left(0\right)+\underset{i=1}{\overset{p}{.Math.}}{?}_{k,l}{U}_{n,j}\left({?}_{n,j}^{*}\right){\widehat{?}}_{n,i}\\ ={?}_k{U}_{n,j}\left(0\right)+Q_{k,n,j}^r{\widehat{?}}_n\end{array}$$\mathrm{where}{Q}_{k,n,j}^T:=\left({?}_{k,1}{U}_{n,j}\left({?}_{n,j}^{*}\right),.Math.,{?}_{k,p}{U}_{n,j}\left({?}_{n,j}^{*}\right)\right),$

where ?*.sub.n,j lies along the line segment connecting {circumflex over (?)}.sub.n and 0. Writing this identity in matrix notation, we have

U.sub.n({circumflex over (?)}.sub.n)?U.sub.n(0)=Q.sub.n({circumflex over (?)}.sub.n) where (Q.sub.n(?)).sub.k,j=Q.sub.n,k,j.sup.T?.

[0109] Let ??(0,1) and ?>0 be given, choose ??(0, ?/K ?{square root over (p)}) so that B.sub.???.sub.0, and let n.sub.2 be such that for all n?n.sub.2, with probability at least 1??, |?.sub.k,lU.sub.n(?)|?K for all 1?k,l?p and ?{circumflex over (?)}.sub.n???.

[0110] Then, for n?n.sub.2 with probability at least 1?? we have

[00013]$.Math.Q_{n,k,j}^T{\widehat{?}}_n.Math.?K\sqrt{p}?<?,\mathrm{orequivalently},{.Math.U_{n^{}}\left({\widehat{?}}_n\right)-U_{n^{}}\left(0\right).Math.}_{?}<?$

where ?A?.sub.?=max.sub.i,j|A.sub.i,j| for A?.sup.p?p. The claim follows, since ? and ? are arbitrary, and U.sub.n(0).fwdarw..sub.p? by assumption. Our next result provides conditions under which a consistent estimator sequence, properly centered and scaled, converges in distribution.

[0111] Theorem 2.2 Suppose the sequence of solutions {circumflex over (?)}.sub.n, n?1 to (7) is consistent for ?.sub.0, that (12) and the second condition of (11) hold for some sequence a.sub.n, n?1 of real numbers, that the matrix ? in (11) is non-singular and that U.sub.n(?) is twice continuously differentiable in an open set ?.sub.0?? containing ?.sub.0. Further, let b.sub.n be a sequence of real numbers such that for some random variable Y,

[00014]$\begin{array}{cc}{b}_n{U}_n\left({?}_0\right){.fwdarw.}_{d}Y.& \left(16\right)\end{array}$$\mathrm{Then}$$\frac{b_n}{a_n}\left({\widehat{?}}_n-{?}_0\right){.fwdarw.}_d-{?}^{-1}Y.$

[0112] Proof: As in the proof of Theorem 2.1, by replacing a.sub.n.sub.n by .sub.n we may without loss of generality take a.sub.n=1, and also as done there, take ?.sub.0=0. Since a limit in distribution does not depend on events of vanishingly small probability, by the consistency of {circumflex over (?)}.sub.n and (12) we may assume that for each n, sufficiently large, that {circumflex over (?)}.sub.n??.sub.0, and for some K that |?.sub.k,jU.sub.n(?)|?K for all 1?j, k?p and ???.sub.0. For such n the expansion (10) holds, and substituting {circumflex over (?)}.sub.n for ? and using U.sub.n({circumflex over (?)}.sub.n)=0 yields

[00015]$-U_n\left(0\right)=\left(U_{n^{}}\left(0\right)+{?}_n\right){\widehat{?}}_n:={?}_n{\widehat{?}}_n\mathrm{where}$${\left({?}_n\right)}_{j,l}=\frac{1}{2}\underset{k=1}{\overset{p}{.Math.}}{\widehat{?}}_{n,k}{?}_{k,l}{U}_{n,j}\left({?}_{n,j}^{*}\right).$

[0113] By the Cauchy-Schwarz inequality,

[00016]$.Math.{\left({?}_n\right)}_{j,l}.Math.?\frac{K\sqrt{p}}{2}.Math.{\widehat{?}}_n.Math.{.fwdarw.}_{p}0.$

[0114] Hence ?.sub.n.fwdarw..sub.p so that ?.sub.n.sup.?1 exists with probability tending to 1, and converges in probability to ?.sup.?1. Now using (16), Slutsky's theorem, on an event of probability tending to one as n tends to infinity,

b.sub.n{circumflex over (?)}.sub.n=?.sub.n.sup.?1(b.sub.n?.sub.n{circumflex over (?)}.sub.n)=??.sub.n.sup.?1(b.sub.nU.sub.n(0)).fwdarw..sub.d??.sup.?1Y.

[0115] In the most common case the distributional convergence in (16) is to the normal, and shown by applying the Central Limit Theorem to a sum of independent random vectors. This situation is illustrated in the following lemma, in which we include distributional limits that may have covariance matrices of less than full rank. For a given vector ? and non-negative definite matrix ?,

we say X?N(?, ?) when E[e.sup.t.sup.T.sup.X]=exp(?t.sup.T?t+t.sup.T?).

[0116] In particular, in one dimension N(?, 0) is unit mass at ?.

[0117] Lemma 2.1 Let =1,2, . . . be a sequence of arbitrary index sets satisfying ||.fwdarw.? as .fwdarw.?, and let {, a?} be a collection of .sup.d valued independent, mean zero random vectors such that for some matrix ? and some ?>0

[00017]$\begin{array}{cc}\underset{?.fwdarw.?}{\lim }{.Math.}_{a?{?}_{?}}\mathrm{Var}\left(X_{?,a}\right)=?\mathrm{and}\underset{?.fwdarw.?}{\lim }{.Math.}_{a?{?}_{?}}?{.Math.X_{?,a}.Math.}^{2+?}=0.& \left(17\right)\end{array}$$\mathrm{Then}{S}_{?}=\underset{a?{?}_{?}}{.Math.}{X}_{?,a}\mathrm{satisfies}{S}_{?}.fwdarw.N\left(0,?\right)\mathrm{as}?.fwdarw.?.$

[0118] Proof: We first prove the result in . By the Lindeberg theorem, (e.g. Theorem 3.4.5, [Durrett, 2019]) if for all ?1 the random variables {, a?.sub.i} are independent, mean zero, and satisfy

[00018]$\begin{array}{cc}\underset{?.fwdarw.?}{\lim }{.Math.}_{a?{?}_{?}}\mathrm{Var}\left(X_{?,a}\right)={?}^2>0,\mathrm{and}\mathrm{for}\mathrm{all}?>0& \left(18\right)\end{array}$$\underset{?.fwdarw.?}{\lim }{.Math.}_{a?{?}_{?}}.Math.X_{?,a}^{2}1\left(.Math.X_{?,a}.Math.??\right)]=0,$

then .fwdarw..sub.dN(0, ?.sup.2) where =X.sub.n,i. In the second condition in (17) implies the second condition in (18), as for any ?>0, with p=1+?/2 and q=1+2/?, using H?lder's inequality followed by Markov's,

[00019]$E\left[X_{?,a}^{2}1\left(.Math.X_{?,a}.Math.??\right)\right]?{E\left[X_{?,a}^{2p}\right]}^{1/p}{P\left(.Math.X_{?,a}.Math.??\right)}^{1/q}?{E\left[X_{?,a}^{2p}\right]}^{1/p}{\left(\frac{E\left[X_{?,a}^{2p}\right]}{{?}^{2p}}\right)}^{1/p}=\frac{E\left[X_{?,a}^{2p}\right]}{{?}^{2p/q}}=\frac{E\left[X_{?,a}^{2+?}\right]}{{?}^{2p/q}}.$

[0119] Hence, the claim holds in when the limiting variance is positive. When this limit is zero, Chebyshev's inequality yields that .fwdarw..sub.p0, and hence converges as well to zero in distribution, which is the normal distribution with mean and variance 0. Hence the conclusion of the lemma holds for d=1.

[0120] In general, given a collection of random vectors satisfying the given hypotheses, taking v to be of norm 1, the variables =v.sup.T for a? are independent and mean zero for each , and satisfy the first condition of (17) with ? replaced by v.sup.T?v, and the second condition of (17) by virtue of this condition holding by assumption for the vector array . and that ||.sup.2+?=|v.sup.T|.sup.2+????.sup.2+?. As the claim holds in d=1 for linear combinations given by any v of norm 1, the general result follows by the Cramer-Wold device.

[0121] Examples (Section 2.3). In the section we demonstrate the scope of the results in Section 2.2 by presenting two applications, one to least squares and the other to maximum likelihood.

[0122] The following lemma, a direct application of the dominated convergence theorem, is used to handle the technical matter of interchanges between integration and differentiation with respect to ????.sup.p.

[0123] Lemma 2.2 Let ?:.sup.m??.fwdarw. be differentiable with respect to ? in an open set ?.sub.0??, and suppose that there exists g:.sup.m.fwdarw. such that

[00020]$.Math.\frac{?}{{?}_{?}}f\left(x;?\right).Math.?g\left(x\right)\mathrm{for}\mathrm{all}??{?}_0\mathrm{and}{?}_{{?}^m}g\left(x\right)\mathrm{dx}<?.$

[0124] Then for all ???.sub.0,

[00021]$\begin{array}{cc}\frac{?}{{?}_{?}}{?}_{{?}^m}f\left(x;?\right)\mathrm{dx}={?}_{{?}^m}\frac{?}{{?}_{?}}f\left(x;?\right)\mathrm{dx}.& \left(19\right)\end{array}$

[0125] Example 2.1 Least squares estimation. Suppose we observe

y.sub.i=?(x.sub.i, ?.sub.0)+?.sub.i i=1, . . . , n

where ?(x.sub.i, ?), ???? is some specified parametric family of functions; we take a one dimensional parameter here for simplicity. We estimate ?.sub.0 via least squares, minimizing

[00022]$J_n\left(?,x^{\left(n\right)}\right)=\frac{1}{2n}\underset{i=1}{\overset{n}{.Math.}}{\left(f\left(x_i,?\right)-y_i\right)}^2=\frac{1}{2n}\underset{i=1}{\overset{n}{.Math.}}{\left(f\left(x_i,?\right)-f\left(x_i,{?}_0\right)-{?}_i\right)}^2.$

[0126] We assume that ?(x, ?) has three continuous derivatives with respect to ? that are uniformly bounded, say by K, over some open subset ?.sub.0 of ? that contains ?.sub.0, and that ?.sub.1, ?.sub.2, . . . are independent random variable distributed as ?, a mean zero, variance ?.sup.2 random variable E|?|.sup.2+?=?.sup.2+?<? for some ?>0.

[0127] Taking one derivative with respect to ?, we obtain the estimating equation .sub.n(?)=0 where

[00023]$\begin{array}{cc}{?}_n\left(?\right)=\frac{1}{n}\underset{i=1}{\overset{n}{.Math.}}\left(f\left(x_i,?\right)-f\left(x_i,{?}_0\right)-{?}_i\right){?}_{?}f\left(x_i,?\right)& \left(20\right)\end{array}$$\mathrm{so}\mathrm{in}\mathrm{particular}{?}_n\left({?}_0\right)=-\frac{1}{n}{.Math.}_{i=1}^n{?}_i{?}_{?}f\left(x_i,{?}_0\right).$

[0128] The first condition of (11) of Theorem 2.1 is satisfied with a.sub.n=1, as the errors have zero mean, are uncorrelated and have uniformly bounded variances, implying that E.sub.?.sub.0[.sub.n(?.sub.0)]=0 and Var.sub.?.sub.0[.sub.n(?.sub.0)].fwdarw.0. Regarding the second condition of (11) taking another derivative, we obtain

[00024]$\begin{array}{cc}\begin{array}{c}{?}_{n^{}}\left({?}_0\right)=\frac{1}{n}\underset{i=1}{\overset{n}{.Math.}}\left({\left({?}_{?}f\left(x_i,?\right)\right)}^2+\left(f\left(x_i,?\right)-f\left(x_i,{?}_0\right)-{?}_i\right){?}_{?}^{2}f\left(x_i,?\right)\right)\\ =\frac{1}{n}{.Math.}_{i=1}^n{\left({?}_{?}f\left(x_i,{?}_0\right)\right)}^2-\frac{1}{n}{.Math.}_{i=1}^n{?}_i{?}_{?}^{2}f\left(x_i,{?}_0\right).\end{array}& \left(21\right)\end{array}$

[0129] Arguing as for (20), the second sum tends to zero in probability. If we now take x.sub.i, i=1,2, . . . to be independent random vectors distributed as some x, then the law of large numbers yields that

[00025]$\begin{array}{cc}\frac{1}{n}{.Math.}_{i=1}^n{\left({?}_{?}f\left(x_i,{?}_0\right)\right)}^2{.fwdarw.}_p?={E_{{?}_0}\left({?}_{?}f\left(x_i,{?}_0\right)\right)}^2,& \left(22\right)\end{array}$

showing the second condition of (11), and this limit will be positive when ?.sub.??(x, ?.sub.0) is a non-degenerate random variable, thus verifying the final condition in (11) in that case.

[0130] It is easy to see that taking another derivative in (21) yields an average of functions that are bounded over ?.sub.0, plus a weighted average of the error variables, each one multiplied by some bounded function. As the second weighted average can be seen to be bounded in probability by applying reasoning similar to that used for the score .sub.n(?.sub.0), condition (12) holds.

[0131] The only remaining verification needed to invoke Theorem 2.2 is to show the properly scaled score at ?.sub.0 has a limiting distribution. Taking b.sub.n=?{square root over (n)}, we have

[00026]$\mathrm{Var}\left(\frac{1}{\sqrt{n}}{?}_n\left({?}_0\right)\right).fwdarw.{?}^2?,$

by (22), and in addition using the representation of .sub.n(?.sub.0) from (20),

[00027]$\underset{i=1}{\overset{n}{.Math.}}?{.Math.\frac{{?}_i{?}_{?}f\left(x_i,{?}_0\right)}{\sqrt{n}}.Math.}^{2+?}?K^{2+?}{n}^{-?/2}{?}^{2+?}.fwdarw.0.$

[0132] Hence, invoking Lemma 2.1, for any consistent sequence of roots,

?{square root over (n)}({circumflex over (?)}.sub.0??.sub.0).fwdarw..sub.dN(0, ?.sup.2?.sup.?1).

[0133] Example 2.2 Maximum likelihood. Let p(x, ?), ???.sub.0 be a family of density functions for ??.sup.p, and for some ?.sub.0??, let X.sub.1, . . . , X.sub.n be independent random vectors with density p(x, ?.sub.0). Let p(x, ?) be three times continuously differentiable in ? with the first two derivatives of p(x, ?), and the third derivative of q(x,?)=log p(x,?), dominated by an integrable function in some neighborhood ?.sub.0 of ?.sub.0. Assume further that the Fisher information matrix at ?.sub.0 is positive definite.

[0134] The maximum likelihood estimate of ?.sub.0 is obtained by maximizing the log likelihood of the data, and hence given by a solution to the estimating equation (7) with

[00028]${?}_n\left(?\right)=\frac{1}{n}\underset{i=1}{\overset{n}{.Math.}}\frac{{?}_{?}p\left(X_i,?\right)}{p\left(X_i,?\right)}$

[0135] By Lemma 2.2, for ???.sub.0 we have

.sub.?[?.sub.? log p(X, ?)]=?.sub.?p(x, ?)dx=?.sub.?p(x, ?)dx=0,(23)

and likewise that the Fisher information I(?) satisfies

l(?=?.sub.?[?.sub.?.sup.2log p(X, ?)]=Var.sub.?(?.sub.?log p(X, ?)).

[0136] Hence, by the law of large numbers the first two conditions of (11) are satisfied with a.sub.n=1 and ?=(?.sub.0), and the last holds by our assumption on the Fisher information. Next we show (12) is satisfied. Writing ?.sub.j short for ?.sub.?.sub.j, we may write

[00029]${?}_n\left(?\right)={?}_n\left(?\right)={?}_{?}q\left(X_i,?\right)\mathrm{andhence}{?}_{k,l}{?}_{n,j}\left(?\right)=\frac{1}{n}\underset{i=1}{\overset{n}{.Math.}}{?}_{k,l,j}q\left(X_i,?\right).$

[0137] Condition (12) can be verified by invoking the following uniform strong law of large numbers with h(x, ?) applied to the components ?.sub.k,i,jq(x, ?).

[0138] Theorem 2.3 Let ? be a compact metric space and ? a space on which a probability distribution F is defined. Let h(x, ?) be measurable in x for each ??? and continuous in ? for almost every x. Assume there exists K(x) such that E[K(X)]<? and |h(x, ?)|?K(x) for all x and ?. Then, with m(?)=E[h(X, ?)].

[00030]$P\left(\underset{n.fwdarw.?}{\lim }\underset{???}{\sup }.Math.\frac{1}{n}\underset{i=1}{\overset{n}{.Math.}}h\left(X_i,?\right)-m\left(?\right).Math.=0\right)=1,$

where X.sub.1, X.sub.2, . . . are independent with distribution F.

[0139] Lastly, under the given assumptions, the classical central limit theorem yields

?{square root over (n)}.sub.n(?.sub.0).fwdarw..sub.d(0, I(?.sub.0))

so that, via Theorem 2.2.

?{square root over (n)}({circumflex over (?)}.sub.0??.sub.0).fwdarw..sub.d(0, I(?.sub.0).sup.?1).

[0140] For the exponential family

p(x; ?)=h(x)exp(?(?)T(x)?A(?)) we have q(x; ?) =log h(x)+?(?)T(x)?A(?).

Hence, the needed conditions are satisfied if A(?) and ?(?) have three bounded derivatives in some neighborhood of ?.sub.0, and E.sub.?.sub.0[T(X)] exists.

[0141] Application to a diffusion equation model (Section 3). To more fully specify the output function of the diffusion model arising from PDE as described herein, consider the parameter space

={(q.sub.1, q.sub.2)?.sup.2:q.sub.2>0},(24)

and for given matrices D, E?.sup.k?k, a vector F?.sup.k and q?, recall from (5) that

A=A(q)=q.sub.1D+E, B=B(q)=q.sub.2F,(25)

and that the TAC at time t is given by

?.sub.?(t; q)=?.sub.0.sup.tCe.sup.A(t?s)B?(s)ds,(26)

where C.sup.T?.sup.k, and ?(s) is the BrAC/BAC at time s. Though our methods work in the given generality, in the physics based model the matrix A will have eigenvalues with negative real parts, and q.sub.1 will be strictly positive. The dependence of ? on A, B, C, ? or q may be dropped in the following for ease of notation, or included to emphasize some particular feature of interest.

[0142] Consider an individual whose data has been collected over i=1, . . . , n drinking sessions, where the BrAC curve ?.sub.i for episode i is integrable on [0, T.sub.i], and for some q.sub.0? and m.sub.i observations of TAC plus a mean zero error

y.sub.ij=?.sub.?.sub.i(t.sub.i,j; q.sub.0)+?.sub.i,j,(27)

are taken at the times 0?t.sub.i,1? . . . ?t.sub.i,j.sub.i?T.sub.i?T, for some T>0. For notational simplicity we may suppress some of the parameters in (27), for instance, denoting ?.sub.?.sub.i(t.sub.i,j; q) by ?.sub.ij(q), say. We encode the observation times of episode i as the probability measure putting mass 1/m.sub.i on each observation time, and form the vector of probability measures v.sub.n=(v.sub.1,m.sub.1, . . . , v.sub.n,m.sub.n). When n=1, that is, for the case of a single episode, we drop the index l.

[0143] For asymptotics, we consider a sequence of experiments indexed by =1,2, . . . , where n and m=(m.sub.1, . . . , m.sub.n) may depend on , and hence we may index using in place of n, m, though this dependence may at times be suppressed in the notation. In the case of a single drinking episode, that is, when n=1, we let =m. For consistency and asymptotic normality, we require that

?.sub.i=1.sup.nm.sub.i.fwdarw.? as .fwdarw.?.(28)

[0144] In the special case where the number of observations m.sub.i for each n equals a constant m, the requirement (28) becomes nm.fwdarw.?, and in the sub-case of a single drinking episode. that m.fwdarw.?.

[0145] Recall that a sequence of measures {v.sub.k, k?1} on is said to converge weakly to a measure v if

[00031]$\underset{k.fwdarw.?}{\min }{?}_{?}g\left(u\right){\mathrm{dv}}_k={?}_{?}g\left(u\right)\mathrm{dv}$$\mathrm{for}\mathrm{all}\mathrm{bounded}\mathrm{continuous}\mathrm{functions}g:?.fwdarw.?.$

[0146] Any sequence {v.sub.k, k?1} of probability measures whose supports are contained in a bounded set is tight, and hence when the weak limit v exists it will also be a probability measure, and its support also so contained.

[0147] There are two special cases of note for the sequence of measures {v.sub.k, k?1}. One is where the distances between consecutive observation times on [0, T] are constant; in this case, the weak limit is the uniform probability measure on [0, T]. A second case is when the observation times are chosen independently according to the probability measure v supported on [0, T]; in this case, the weak limit in probability is ?.

[0148] Let the gradient of (q) be denoted

[00032]$\begin{array}{cc}{?}_q{J}_{?}\left(q\right)=\left(\begin{array}{c}{?}_1{J}_{?}\left(q\right)\\ {?}_2{J}_{?}\left(q\right)\end{array}\right)\mathrm{and}\mathrm{let}& \left(29\right)\end{array}$$g_{?}\left(u;q\right)={?}_q{f}_{?}\left(u;q\right){?}_q{f_{?}\left(u;q\right)}^{?}.$

[0149] We apply the methods developed herein to the least squares estimator achieved as a solution to

(q)=0 where (q)=?.sub.q(q),(30)

where (q) is given by the sum of squares in (6). For i?{1,2}, we continue to let ?.sub.i denote taking the partial derivative with respect to q.sub.i; this notation will extend in the natural way to denote higher order, and mixed partial derivatives. Theorem 3.1 below gives conditions under which the least squares estimate is consistent and has a limiting, asymptotically normal distribution, and as well provides the form of the limiting covariance matrix. Theorem 3.1 is an immediate consequence of Theorems 3.2 and 3.3. that verify the conditions of Theorems 2.1 and 2.2 in the previous section.

[0150] To set the stage for the statements and proofs of our results, we note that when v.sub.n, m?1 is the discrete probability measure giving equal weight to the times t.sub.m,1, . . . , t.sub.m,m in [0, T], then for any continuous function h:[0, T].fwdarw., when v.sub.m converges weakly to v, we have

[00033]$\begin{array}{cc}\frac{1}{m}{.Math.}_{j=1}^{m}h\left(t_{m,j}\right)={?}_0^{T}h\left(u\right){\mathrm{dv}}_m.fwdarw.{?}_0^{T}h\left(u\right)\mathrm{dv}.& \left(31\right)\end{array}$

[0151] By considering components, the same relations hold when h continuously maps [0, T] to the space of matrices of some fixed dimension. For a given BrAC curve ?, of particular interest is the .sup.2?2 valued matrix function g.sub.? in (29) that determines, via (q), the limiting covariance matrix ? of our q parameter estimate.

[0152] We consider two special cases where the existence of the limit ? is guaranteed. For a single drinking episode, that is, when n=1, when v.sub.m converges weakly to v, due to the continuity of elements of g.sub.?(u) as shown in Lemma 3.3, we have, as m.fwdarw.?,

[00034]$\begin{array}{cc}\begin{array}{c}{?}_m={?}_0^T{g}_{?}\left(u\right){\mathrm{dv}}_m.fwdarw.{?}_0^T{g}_{?}\left(u\right)\mathrm{dv}\\ =\left(\begin{array}{cc}{?}_0^T{\left({?}_1{f}_{?}\left(u\right)\right)}^2\mathrm{dv}& \frac{1}{q_{0.2}}{?}_0^T{f}_{?}\left(u\right){?}_1{f}_{?}\left(u\right)\mathrm{dv}\\ \frac{1}{q_{0.2}}{?}_0^T{f}_{?}\left(u\right){?}_1{f}_{?}\left(u\right)\mathrm{dv}& \frac{1}{q_{0.2}^2}{?}_0^T{f_{?}\left(u\right)}^2\mathrm{dv}\end{array}\right)\\ =?\end{array}& \left(32\right)\end{array}$

In particular, v will be the uniform probability measure on [0, T] when the number m of sampling times tend to infinity, and the consecutive distances between them are equal.

[0153] For another case, consider a situation where the data from n drinking episodes are independent and identically distributed from replicates of the error distribution and canonical M, T, ?, v, where M is the distribution of m.sub.i, 1?i?n, making the summands in (33) i.i.d. When T<?a.s. Lemma 3.3 shows that the integrals in (33) are uniformly bounded, and one can show that as n.fwdarw.?.

[00035]${?}_n{.fwdarw.}_p?=E\left[\frac{M}{E\left[M\right]}\left(\begin{array}{cc}{?}_0^T{\left({?}_1{f}_{?}\left(u\right)\right)}^2\mathrm{dv}& \frac{1}{q_{0.2}}{?}_0^T{f}_{?}\left(u\right){?}_1{f}_{?}\left(u\right)\mathrm{dv}\\ \frac{1}{q_{0.2}}{?}_0^T{f}_{?}\left(u\right){?}_1{f}_{?}\left(u\right)\mathrm{dv}& \frac{1}{q_{0.2}^2}{?}_0^T{f_{?}\left(u\right)}^2\mathrm{dv}\end{array}\right)\right],$

where the expectation is taken over M, T, ? and v, whenever the expectation on the right hand side exists. We now present our main result regarding the least squares estimator for the diffusion model.

[0154] Theorem 3.1 Suppose the errors ?.sub.i,j, 1?i?n. 1?j?m.sub.i in model (27) are mean zero, uncorrelated and have constant positive variance ?.sup.2. With ?.sub.i and v the BrAC curve and the empirical measure of the observation times for episode i=1, . . . , n, we assume the existence of the limit

[00036]$\begin{array}{cc}?=\underset{?.fwdarw.?}{\lim }{?}_{?}\mathrm{where}& \left(33\right)\end{array}$${?}_{?}={.Math.}_{i=1}^n\frac{m_i}{{.Math.}_{k=1}^n{m}_k}{?}_0^{T_i}{?}_q{f}_{{?}_i}\left(u;q_0\right){?}_q{f_{{?}_i}\left(u;q_0\right)}^{?}{\mathrm{dv}}_i.$

that ? is positive definite, and that (28) holds. Then there exists a consistent sequence of solutions to the estimating equation (q)=0.

[0155] If in addition the errors ?.sub.i,j are i.i.d., and for some ?>0 satisfy E|?.sub.i,j|.sup.2+?=?.sup.2+?<?, then, along any such consistent sequence ,

[00037]$\begin{array}{cc}\sqrt{m}\left({\hat{q}}_{?}-q_0\right){.fwdarw.}_d?\left(0,{?}^2{?}^{-1}\right)\mathrm{where}m=\underset{i=1}{\overset{n}{.Math.}}{m}_i\mathrm{and}& \left(34\right)\end{array}$${\widehat{?}}_{?}^2{.fwdarw.}_p{?}^2\mathrm{where}{\widehat{?}}_{?}^2=\frac{1}{m}{.Math.}_{i=1}^n{.Math.}_{j=1}^{m_i}{\left(y_{\mathrm{ij}}-f_{\mathrm{ij}}\left({\hat{q}}_{?}\right)\right)}^2.$

[0156] When the errors ?.sub.i,j in (27) are Gaussian, then the least squares estimate that minimizes the sum of squares (6) is also maximum likelihood. In this case the contribution to the Fisher information from the single observation in (27) is obtained by taking the covariance matrix of the gradient of the log of the density of the observation,

[00038]$\mathrm{Var}\left({?}_q\log p\left(y_{\mathrm{ij}};f_{\mathrm{ij}}\right)\right)=\mathrm{Var}\left({?}_q\log \left(\frac{1}{\sqrt{2?}?}\exp \left(-\frac{1}{2{?}^2}{\left(y_{\mathrm{ij}}-f_{\mathrm{ij}}\right)}^2\right)\right)\right)=\mathrm{Var}\left({?}_q\left(-\frac{1}{2{?}^2}{\left(y_{\mathrm{ij}}-f_{\mathrm{ij}}\right)}^2\right)\right)=\mathrm{Var}\left(\frac{{?}_{\mathrm{ij}}}{{?}^2}{?}_q{f}_{\mathrm{ij}}\right)=\frac{1}{{?}^2}{?}_q{f}_{\mathrm{ij}}{?}_q{f}_{\mathrm{ij}}^{?}.$

[0157] Summing over the observation times yields m/?.sup.2 as in (33), hence taking the limit and comparing with the asymptotic variance obtained we see that for normal errors the least squares estimate of q achieves the lower bound of the information inequality in an asymptotic sense.

[0158] Before proceeding, we must verify the smoothness of the derivatives of ?.sub.?(t; q) in (26) with respect to q=(q.sub.1, q.sub.2). Because of the form of the dependence of the matrix A on q.sub.1 as given in (25), to differentiate ? with respect to q.sub.1 we will need to consider directional derivatives of matrix exponentials. For square matrices W and V of the same dimension and u?, define the first derivative of e.sup.uW in direction V by

[00039]${?}_V^1\left(u,W\right)=\underset{h.fwdarw.0}{\lim }\frac{\exp \left(u\left(W+\mathrm{hV}\right)\right)-\exp \left(\mathrm{uW}\right)}{h}.$

[0159] We define higher order derivatives .sub.V.sup.k(u, W), k?0 in the natural way, with k=0 returning e.sup.uW. Now with A=q.sub.1D+E as in (25), we may represent the partial derivative with respect to q.sub.1 of e.sup.uA as

[00040]${?}_1{e}^{\mathrm{uA}}={?}_1{e}^{u\left(q_{1}D+E\right)}=\underset{h.fwdarw.0}{\lim }\frac{e^{u\left(\left(q_1+h\right)D+E\right)}-e^{u\left(q_{1}D+E\right)}}{h}=\underset{h.fwdarw.0}{\lim }\frac{e^{u\left(A+\mathrm{hD}\right)}-e^{\mathrm{uA}}}{h}$$={?}_D^1\left(u,A\right),$

and extending to higher order derivatives we obtain

?.sub.1.sup.n(e.sup.(q.sup.1.sup.D+E)?)=.sub.D.sup.n(u, A).(35)

[0160] For any n?0, letting B.sub.n be the (n+1)?(n+1) block matrix given by

[00041]$\begin{array}{cc}{B}_n=\left[\begin{array}{ccccc}W& V& 0& .Math.& 0\\ 0& W& V& .Math.& 0\\ .Math.& .Math.& .Math.& .Math.& .Math.\\ 0& 0& .Math.& 0& W\\ & & & & \end{array}\right],& \left(36\right)\end{array}$

[0161] A known theorem provides that

[00042]$\begin{array}{cc}{e}^{{\mathrm{uB}}_n}=\left[\begin{array}{ccccc}{e}^{\mathrm{uW}}& \frac{{?}_V^1\left(u,W\right)}{1!}& \frac{{?}_V^2\left(u,W\right)}{2!}& .Math.& \frac{{?}_V^n\left(u,W\right)}{n!}\\ 0& e^{\mathrm{uW}}& \frac{{?}_V^1\left(u,W\right)}{1!}& .Math.& \frac{{?}_V^{n-1}\left(u,W\right)}{\left(n-1\right)!}\\ .Math.& .Math.& .Math.& .Math.& .Math.\\ 0& 0& .Math.& 0& e^{\mathrm{uW}}\\ & & & & \end{array}\right].& \left(37\right)\end{array}$

[0162] We now apply (37) to obtain bounds on higher order derivatives of the matrix exponential e.sup.uA with respect to q.sub.1.

[0163] Lemma 3.1 Let W and V be square matrices of the same dimension. Then for all n?0 the directional derivative .sub.V.sup.n(u, W) is analytic in u on and satisfies the bound

?.sub.V.sup.n(u, W)??n!?e.sup.uB.sup.n? for all u?,(38)

where B.sub.n is given by (36). For all n?0, q.sub.1? and A=q,.sub.1D+E, the partial derivative ?.sub.1.sup.ne.sup.Au exists, is analytic in q.sub.1 and satisfies ??.sub.1.sup.ne.sup.uA??n!e.sup.u?B.sup.n? where B.sub.n is given by (36) with W=A and V=D.

[0164] Proof: As the left hand side e.sup.uB.sup.n of (37) is analytic in each component, the matrix on the right hand side must also be analytic, thus yielding the first claim. Next, for F the submatrix obtained by taking row and column indices i, j of a given matrix E, applying an alternate form for the spectral norm in the first equality, we have

[00043]$.Math.F.Math.=\underset{.Math.y.Math.=1,.Math.x.Math.=1}{\sup }{y}^{?}\mathrm{Fx}?\underset{.Math.u.Math.=1,.Math.v.Math.=1}{\sup }{u}^{?}\mathrm{Ev}=.Math.E.Math.,$

as any value over which the first supremum is taken can be achieved in the second by padding x and y with zeros in coordinates that are not in i and j, respectively. Hence, inequality (38) now follows from (37). The remaining claims now follow in light of (35).

[0165] We require the following result to handle the derivatives of matrix products. For k?0, Q as in (24), we say a matrix M depending on (q, u)?Q? is k-smooth if for any 0?j.sub.1, j.sub.2?k, the mixed partials ?.sub.1.sup.j.sup.1?.sub.2.sup.j.sup.2M exist and are continuous for q?, and for any bounded subsets ? and I?,

[00044]$\underset{\left(q,u\right)?D?i,0?j_1,j_2?k}{\sup }.Math.{?}_1^{J_1}{?}_2^{J_2}M.Math.<?.$

We say M is smooth if it is k-smooth for all k?0.

[0166] Lemma 3.2 Let M.sub.i, i=1, . . . , d be matrices having dimensions such that we may form the product

[00045]$M=\underset{i=1}{\overset{d}{.Math.}}{M}_i.$

If M.sub.1, . . . , M.sub.d are k-smooth then so is M.

[0167] Proof: The proof follows directly from the multivariate Leibniz rule that expresses the derivative ?.sub.1.sup.j.sup.1?.sub.2.sup.j.sup.2M.sub.i for 0?j.sub.1, j.sub.2?k as a finite linear combination of products of derivatives of M.sub.i, each one with order no greater than k, and recalling that for conformable matrices ?AB???A??B?.

[0168] The next lemma provides us with additional smoothness estimates, and the forms of derivatives that later appear.

[0169] Lemma 3.3 For all u? the matrix function e.sup.AuB is smooth in q. If ?(?) is integrable on [0, T], then ?.sub.?(t; q) as in (26) is smooth in q, continuous for t?[0, T] and satisfies

|?.sub.?(t; q)|?q.sub.2e.sup.T(q.sup.1.sup.?D?+?E?)???.sub.1 and ?.sub.1.sup.j.sup.1?.sub.2.sup.j.sup.2?.sub.?(t; q)=?.sub.0.sup.t?.sub.1.sup.j.sup.1?.sub.2.sup.j.sup.2(Ce.sup.A(t?s)B)?(s)ds.

[0170] For q?, and

?.sub.1(e.sup.AuB)=?.sub.1(e.sup.Au)B and ?.sub.2(e.sup.AuB)=q.sub.2.sup.?1e.sup.AuB.(39)

[0171] Proof: That e.sup.Au is smooth follows from Lemma 3.1, and one easily verifies the smoothness of B directly from (25); hence, the product is smooth by Lemma 3.2. Differentiation under the integral is then justified by the dominated convergence theorem, from which the smoothness of ?.sub.?(t; q) in q then follows; continuity for t?[0, T] follows immediately from the integral representation (26). The claims in (39) follow by recalling that B=q.sub.2F.

[0172] We now begin to verify the conditions of Theorems 2.1 and 2.2.

[0173] Theorem 3.2 Suppose the errors ?.sub.i,j, 1?i?n, 1?j?m.sub.i in model (27) are mean zero, uncorrelated and have constant positive variance ?.sup.2. Assume in addition that the limit ? in (33) exists and is positive definite, and that (28) holds. Then with given by (??) and (6), the hypotheses of Theorem 2. 1 are satisfied with ? as in (33), a.sub.n=1 and any bounded neighborhood ?.sub.0? of q.sub.0.

[0174] Proof: Let ?.sub.0 be any bounded neighborhood of q.sub.0. By Lemma 3.3, the partial derivatives of ?.sub.ij(q):=?.sub.?(t.sub.i,j; q) of (26) of all orders exist, and are continuous and uniformly bounded over ?.sub.0. Hence is twice continuously differentiable, with uniformly bounded derivatives, over ?.sub.0.

[0175] Now write the score function as

[00046]$\begin{array}{cc}{?}_{?}\left(q\right)={?}_{?,1}\left(q\right)-{?}_{?,2}\left(q\right)& \left(40\right)\end{array}$$\mathrm{where}$${?}_{?,1}\left(q\right)=\frac{1}{{.Math.}_{i=1}^n{m}_i}\underset{i=1}{\overset{n}{.Math.}}\underset{j=1}{\overset{m_i}{.Math.}}{?}_q{f}_{\mathrm{ij}}\left(q\right)\left(f_{\mathrm{ij}}\left(q\right)-f_{\mathrm{ij}}\left(q_0\right)\right)$$\mathrm{and}{?}_{?,2}\left(q\right)=\frac{1}{{.Math.}_{i=1}^n{m}_i}\underset{i=1}{\overset{n}{.Math.}}\underset{j=1}{\overset{m_i}{.Math.}}{?}_q{f}_{\mathrm{ij}}\left(q\right){?}_{\mathrm{ij}}.$

[0176] In particular,

[00047]$\begin{array}{cc}{?}_{?}\left(q_0\right)=-\frac{1}{{.Math.}_{i=1}^n{m}_i}{.Math.}_{i=1}^m{.Math.}_{j=1}^{m_i}{?}_q{f}_{\mathrm{ij}}\left(q_0\right){?}_{\mathrm{ij}}.& \left(41\right)\end{array}$

[0177] Differentiating (40) and evaluating at q.sub.0, we find

(q.sub.0)=(q.sub.0)?(q.sub.0),(42)

where, using (33),

[00048]${?}_{?,1}^{}\left(q_0\right)=\frac{1}{{.Math.}_{i=1}^n{m}_i}\underset{i=1}{\overset{n}{.Math.}}\underset{j=1}{\overset{m_i}{.Math.}}{?}_q{f}_{\mathrm{ij}}\left(q_0\right){?}_q{f_{\mathrm{ij}}\left(q_0\right)}^{?}={?}_l$$\mathrm{and}{?}_{?,2}^{}\left(q_0\right)=\frac{1}{{.Math.}_{i=1}^n{m}_i}\underset{i=1}{\overset{n}{.Math.}}\underset{j=1}{\overset{m_i}{.Math.}}{?}_q^2{f}_{\mathrm{ij}}\left(q_0\right){?}_{\mathrm{ij}}.$

[0178] To show the first condition in (11), note that [.sub.n(q.sub.0)]=0 as the error variables have mean zero. Next, using that the error variables are uncorrelated and have constant variance yields that the covariance matrix =Var((q.sub.0)) is given by

[00049]$\begin{array}{cc}{?}_{?}=\frac{{?}^2}{{\left({.Math.}_{i=1}^n{m}_i\right)}^2}{.Math.}_{i=1}^n{.Math.}_{j=1}^m{?}_q{f}_{\mathrm{ij}}\left(q_0\right){?}_q{f_{\mathrm{ij}}\left(q_0\right)}^{?}=\frac{{?}^2}{{.Math.}_{i=1}^n{m}_i}{?}_{?}.& \left(43\right)\end{array}$

[0179] The claim follows by noting that ?.sub.2?.sub.?(q)=(1/q.sub.2)?.sub.?(q) by Lemma 3.3, and that ??.sub.n?.fwdarw.0 as ?.sub.n.fwdarw.? and ?.sub.im.sub.i.fwdarw.? by assumption. For the second condition in (11), we recognize that .sub.n,1(q.sub.0)=?.sub.n, and can show that the components of .sub.n,2(q.sub.0) have mean zero and variance converging to zero, so that the sum of these two matrices tends to ? in probability as n.fwdarw.?. The matrix ? is positive definite by assumption, so the last condition in (11) holds.

[0180] Lastly, we show that inequality (12) is satisfied. From the decomposition (40) we see that we may write ?.sub.k,lU.sub.n,r(q) as a difference of the form

[00050]$R_n-S_n:=\frac{1}{{.Math.}_{i=1}^n{m}_i}\underset{i=1}{\overset{n}{.Math.}}\underset{j=1}{\overset{m_i}{.Math.}}{g}_{1,\mathrm{ij}}\left(q\right)-\frac{1}{{.Math.}_{i=1}^n{m}_i}\underset{i=1}{\overset{n}{.Math.}}\underset{j=1}{\overset{m_i}{.Math.}}{g}_{2,\mathrm{ij}}\left(q\right){?}_{i,j}$

for some functions g.sub.p,ij,p=1,2, where, by Lemma 3.3 and the product rule, for some K.sub.1

[00051]$\underset{q?Q_0,p?\left\{1,2\right\}}{\sup }.Math.g_{p,\mathrm{ij}}\left(q\right).Math.?K_1.$

[0181] Hence, for the first component,

[00052]$.Math.R_n.Math.=.Math.\frac{1}{{.Math.}_{i=1}^n{m}_i}\underset{i=1}{\overset{n}{.Math.}}\underset{j=1}{\overset{m_i}{.Math.}}{g}_{1,\mathrm{ij}}\left(q\right).Math.?\frac{1}{{.Math.}_{i=1}^n{m}_i}\underset{i=1}{\overset{n}{.Math.}}\underset{j=1}{\overset{m_i}{.Math.}}{K}_1=K_1,$

while for the second component,

[00053]$\mathrm{Var}\left(S_n\right)?\frac{{?}^2}{{\left({.Math.}_{i=1}^n{m}_i\right)}^2}\underset{i=1}{\overset{n}{.Math.}}\underset{j=1}{\overset{m_i}{.Math.}}{K}_1^2?\frac{{?}^2{K}_1^2}{{.Math.}_{i=1}^n{m}_i}.fwdarw.0.$

[0182] Hence, for any ??(0,1), by Chebyshev's inequality, we may pick K.sub.2 such that P(|S.sub.n|?K.sub.2)??/8 for all n?1. Thus, setting K=K.sub.1+K.sub.2, we obtain, for all n?1,

[00054]$P\left(.Math.R_n-S_n.Math.>K,q?Q_0\right)?P\left(.Math.R_n.Math.+.Math.S_n.Math.>K,q?Q_0\right)?P\left(K_1+.Math.S_n.Math.>K_1+K_2,q?Q_0\right)=P\left(.Math.S_n.Math.>K_2,q?Q_0\right)?\frac{?}{8}.$

[0183] The claim now follows by taking a union bound over the eight choices for k, l and r.

[0184] Theorem 3.3 Assume the errors ?.sub.ij, 1?i?n, 1?j?m.sub.i are i.i.d with mean zero, variance ?.sup.2 and for some ?>0 we have E|?.sub.ij|.sup.2+?=?.sup.2+?<?. Assume that (28) holds and that the limit ? as given in (33) exists. Then for .sub.n(q),

[00055]$\begin{array}{ccc}{b}_n{?}_n\left(q_0\right){.fwdarw.}_d?\left(0,{?}^2?\right)& \mathrm{where}& b_n=\sqrt{\underset{i=1}{\overset{n}{.Math.}}{m}_i}.\end{array}$

[0185] Proof: We verify (17) of Lemma 2.1. For the first condition there,

[00056]$\begin{array}{cc}{?}_n\left(q_0\right)=\frac{1}{{.Math.}_{i=1}^n{m}_i}{.Math.}_{i=1}^n{.Math.}_{j=1}^{m_i}\left(\begin{array}{c}{?}_1{f}_{\mathrm{ij}}\left(q_0\right)\\ {?}_2{f}_{\mathrm{ij}}\left(q_0\right)\end{array}\right){?}_{\mathrm{ij}}& \left(44\right)\end{array}$

we see that the mean of b.sub.n.sub.n(q.sub.0) is zero, and by (33)

[00057]$\mathrm{Cov}\left(b_n{?}_n\left(q_0\right)\right)={?}^2\underset{i=1}{\overset{n}{.Math.}}\frac{m_i}{{.Math.}_{k=1}^n{m}_k}{?}_0^{T_i}\left(\begin{array}{cc}{?}_1{f_{\mathrm{ij}}\left(q_0\right)}^2& {?}_1{f}_{\mathrm{ij}}\left(q_0\right){?}_2{f}_{\mathrm{ij}}\left(q_0\right)\\ {?}_1{f}_{\mathrm{ij}}\left(q_0\right){?}_2{f}_{\mathrm{ij}}\left(q_0\right)& {?}_1{f_{\mathrm{ij}}\left(q_0\right)}^2\end{array}\right){\mathrm{dv}}_{i,n}={?}^2{?}_n.fwdarw.{?}^2?.$

[0186] For the second condition of (17), write

[00058]$\begin{array}{ccc}{b}_n?\left(q_0\right)=\underset{i=1}{\overset{n}{.Math.}}\underset{j=1}{\overset{m_i}{.Math.}}{X}_{\mathrm{ij}}& \mathrm{where}& X_{\mathrm{ij}}=-\end{array}\frac{1}{\sqrt{{.Math.}_{i=1}^n{m}_i}}\left(\begin{array}{c}{?}_1{f}_{\mathrm{ij}}\left(q_0\right)\\ {?}_2{f}_{\mathrm{ij}}\left(q_0\right)\end{array}\right){?}_{\mathrm{ij}}.$

[0187] By the assumption |?.sub.ij|.sup.2+???.sup.2+? and Lemma 3.3 there exists C such that

[00059]$\underset{1?i?n,1?j?m_i}{.Math.}E{.Math.X_{\mathrm{ij}}.Math.}^{2+?}?\frac{C{?}^{2+?}}{{\left({.Math.}_{i=1}^n{m}_i\right)}^{1+?/2}},$

which tends to zero by (28).

[0188] We conclude this section with: Proof of Theorem 3.1: Theorems 3.2 and 3.3 show that the hypotheses of Theorems 2.1 and 2.2 are satisfied, yielding the claims for consistency and asymptotic normality. It remains to prove the claims on the consistency of the variance estimator. By (34), and letting m=?.sub.i=1.sup.nm.sub.i, we have

[00060]${\widehat{?}}_n^2=\frac{1}{m}\underset{i=1}{\overset{n}{.Math.}}\underset{j=1}{\overset{m_i}{.Math.}}{\left({?}_{\mathrm{ij}}+f_{\mathrm{ij}}\left(q_0\right)-f_{\mathrm{ij}}\left({\hat{q}}_n\right)\right)}^2=\frac{1}{m}\underset{i=1}{\overset{n}{.Math.}}\underset{j=1}{\overset{m_i}{.Math.}}({?}_{\mathrm{ij}}^2+2{?}_{\mathrm{ij}}(f_{\mathrm{ij}}\left(q_0\right)-f_{\mathrm{ij}}\left({\hat{q}}_n\right)+\left(f_{\mathrm{ij}}\left(q_0\right)-{f_{\mathrm{ij}}\left({\hat{q}}_n\right)}^2\right).$

[0189] The first term tends to ?.sup.2 in probability by the weak law of large numbers. To handle the second term, letting

[00061]$\begin{array}{ccc}R=\frac{1}{m}\underset{i=1}{\overset{n}{.Math.}}\underset{j=1}{\overset{m_i}{.Math.}}.Math.{?}_{\mathrm{ij}}.Math.& \mathrm{we}\mathrm{have}& E\left[R\right]=\end{array}\frac{1}{m}\underset{i=1}{\overset{n}{.Math.}}\underset{j=1}{\overset{m_i}{.Math.}}E.Math.{?}_{\mathrm{ij}}.Math.?\frac{1}{m}\underset{i=1}{\overset{n}{.Math.}}\underset{j=1}{\overset{m_i}{.Math.}}\sqrt{E{?}_{\mathrm{ij}}^2}=?.$

[0190] With B.sub.1 the unit ball centered at q.sub.0, Lemma 3.3 shows that the first derivatives of ?.sub.j(q) are uniformly bounded for (q, t)?B.sub.1?[0, T], that is, there exists some K>0 such that over this set

|?.sub.i j(q.sub.0)??.sub.i j(q)|?K?q.sub.0?q?.(45)

[0191] Let ??(0, K) be arbitrary, F={|q.sub.0?{circumflex over (q)}.sub.n|??/K} and note that

[00062]$\begin{array}{ccc}S=.Math.\frac{1}{m}\underset{i=1}{\overset{n}{.Math.}}\underset{j=1}{\overset{m_i}{.Math.}}{?}_{\mathrm{ij}}(f_{\mathrm{ij}}\left(q_0\right)-f_{\mathrm{ij}}\left({\hat{q}}_n\right).Math.& \mathrm{satisfies}& S{1}_F??R{1}_F.\end{array}$

[0192] By Markov's inequality, for any ?>0,

[00063]$P\left(S??\right)?P\left(S{1}_F??\right)+P\left(F^c\right)?P\left(?R{1}_F??\right)+P\left(F^c\right)?\frac{?E\left[R{1}_F\right]}{?}+P\left(F^c\right)?\frac{?E\left[R\right]}{?}+P\left(F^c\right)?\frac{\mathrm{??}}{?}+P\left(F^c\right).fwdarw.\frac{\mathrm{??}}{?},$

using the non-negativity of R in the fourth inequality, and the consistency of q.sub.n when taking the limit. As ? can be made arbitrarily small we conclude that P(S??).fwdarw.0, and as t is arbitrary, that S.fwdarw..sub.p0.

[0193] Similarly decomposing the third term on the good event where q.sub.n is in B.sub.1, and the complentary event which tends in probability to zero, on the good event, applying the inequality (45), this last term is bounded as

[00064]$\frac{K^2}{m}\underset{i=1}{\overset{n}{.Math.}}\underset{j=1}{\overset{m_i}{.Math.}}{.Math.q_0-{\hat{q}}_n.Math.}^2=K^2{.Math.q_0-{\hat{q}}_n.Math.}^2,$