Computation of nonparametric, mixed effects, maximum likelihood, biosensor data based-estimators for the distributions of random parameters in an abstract parabolic model for 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 was 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 was also established and the efficacy of the approach was demonstrated by applying the scheme to the transdermal transport of alcohol modeled by a random parabolic partial differential equation (PDE). Numerical studies included show that the maximum likelihood estimator is statistically consistent, demonstrated by the convergence of the estimated distribution to the "true" distribution in an example involving simulated data. The algorithm developed was then applied to two datasets collected using two different transdermal alcohol biosensors. Using the leave-one-out cross-validation (LOOCV) method, we found an estimate for the distribution of the random parameters based on a training set. The input from a test drinking episode was 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.