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.

翻译：针对离散时间抽象抛物系统中随机参数的联合概率分布，本文采用非参数方法，在混合效应统计模型的框架下，利用可行测度集上的Prohorov度量结构，证明了最大似然估计的存在性与相合性。同时，建立了计算最大似然估计的有限维逼近方案的理论收敛性结果，并通过将该方案应用于由随机抛物偏微分方程建模的酒精经皮传输过程，验证了该方法的效果。数值研究表明，最大似然估计具有统计相合性：在涉及模拟数据的示例中，观测到估计分布趋近于"真实"分布。随后，将所开发的算法应用于两种不同经皮酒精生物传感器采集的两组数据。通过留一法交叉验证，基于训练集获得了随机参数分布的估计。利用测试饮酒事件的数据输入，将随机参数的不确定性传播至模型输出，并以估计输出信号周围95%误差带的形式量化了该不确定性。