Prohorov Metric-Based Nonparametric Estimation of the Distribution of Random Parameters in Abstract Parabolic Systems with Application to the Transdermal Transport of Alcohol

We consider a Prohorov metric-based nonparametric approach to estimating the probability distribution of a random parameter vector in discrete-time abstract parabolic systems. We establish the existence and consistency of a least squares estimator. We develop a finite-dimensional approximation and convergence theory, and obtain numerical results by applying the nonparametric estimation approach and the finite-dimensional approximation framework to a problem involving an alcohol biosensor, wherein we estimate the probability distribution of random parameters in a parabolic PDE. To show the convergence of the estimated distribution to the "true" distribution, we simulate data from the "true" distribution, apply our algorithm, and obtain the estimated cumulative distribution function. We then use the Markov Chain Monte Carlo Metropolis Algorithm to generate random samples from the estimated distribution, and perform a generalized (2-dimensional) two-sample Kolmogorov-Smirnov test with null hypothesis that our generated random samples from the estimated distribution and generated random samples from the "true" distribution are drawn from the same distribution. We then apply our algorithm to actual human subject data from the alcohol biosensor and observe the behavior of the normalized root-mean-square error (NRMSE) using leave-one-out cross-validation (LOOCV) under different model complexities.