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The existence and consistency of a maximum likelihood estimator for the jointprobability distribution of random parameters in discrete-time abstractparabolic systems are established by taking a nonparametric approach in thecontext of a mixed effects statistical model using a Prohorov metric frameworkon a set of feasible measures. A theoretical convergence result for a finitedimensional approximation scheme for computing the maximum likelihood estimatoris also established and the efficacy of the approach is demonstrated byapplying the scheme to the transdermal transport of alcohol modeled by a randomparabolic PDE. Numerical studies included show that the maximum likelihoodestimator is statistically consistent in that the convergence of the estimateddistribution to the "true" distribution is observed in an example involvingsimulated data. The algorithm developed is then applied to two datasetscollected using two different transdermal alcohol biosensors. Using theleave-one-out cross-validation method, we get an estimate for the distributionof the random parameters based on a training set. The input from a testdrinking episode is then used to quantify the uncertainty propagated from therandom parameters to the output of the model in the form of a 95% error bandsurrounding the estimated output signal.

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