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.

翻译：本文提出一种基于Prohorov度量的非参数方法，用于估计离散时间抽象抛物系统中随机参数向量的概率分布。我们证明了最小二乘估计量的存在性和相合性。发展了有限维逼近与收敛理论，并将该非参数估计方法及有限维逼近框架应用于酒精生物传感器问题，通过估计抛物型偏微分方程中随机参数的概率分布获得了数值结果。为展示估计分布对"真实"分布的收敛性，我们从"真实"分布中模拟数据，应用算法获取估计累积分布函数。继而采用马尔可夫链蒙特卡罗Metropolis算法从估计分布中生成随机样本，进行广义（二维）双样本Kolmogorov-Smirnov检验，其原假设为：从估计分布生成的随机样本与从"真实"分布生成的随机样本来自同一分布。最后将算法应用于酒精生物传感器的实际人体实验数据，通过留一交叉验证法观察不同模型复杂度下归一化均方根误差的行为。