  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.
