Asymptotic Properties of Probability Measure Estimators in a Nonparametric Model

We consider probability measure estimation in a nonparametric model using a leastsquares approach under the Prohorov metric framework. We summarize the computational methods and their convergence results that were developed by our group over the past two decades. New results are presented on the bias and the variance due to the approximation and the pointwise asymptotic normality of the approximated probability measure estimator. We propose use of model selection criterion to balance the bias and the variance, and compare the pointwise confidence band constructed using the asymptotic normality results with that obtained by Monte Carlo simulations.