  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.
