A Nonparametric, Mixed Effect, Maximum Likelihood Estimator for the Distribution of Random Parameters in Discrete-Time Abstract Parabolic Systems with Application to the Transdermal Transport of Alcohol

The existence and consistency of a maximum likelihood estimator for the joint probability distribution of random parameters in discrete-time abstract parabolic systems are established by taking a nonparametric approach in the context of a mixed effects statistical model using a Prohorov metric framework on a set of feasible measures. A theoretical convergence result for a finite dimensional approximation scheme for computing the maximum likelihood estimator is also established and the efficacy of the approach is demonstrated by applying the scheme to the transdermal transport of alcohol modeled by a random parabolic PDE. Numerical studies included show that the maximum likelihood estimator is statistically consistent in that the convergence of the estimated distribution to the "true" distribution is observed in an example involving simulated data. The algorithm developed is then applied to two datasets collected using two different transdermal alcohol biosensors. Using the leave-one-out cross-validation method, we get an estimate for the distribution of the random parameters based on a training set. The input from a test drinking episode is then used to quantify the uncertainty propagated from the random parameters to the output of the model in the form of a 95% error band surrounding the estimated output signal.