Asymptotic Properties of Probability Measure Estimators in a Nonparametric Model

Abstract

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.

Estimation of distributed parameters in permittivity models of composite dielectric materials using reflectance

“…This is especially true for parameter τ : its relative error for the cases of N = 25 and 30 is 0.14 and 0.30 respectively, and it is considerably higher than those obtained with a lower value of N (for N = 5, 10, 15, 20 the mean relative error is 0.06). We remark that the loss of estimation accuracy as N increases is in agreement with the common understanding that for a fixed number of observations the estimation accuracy in general decreases as the number of estimated parameters increases (e.g., see [7,19]). Actually, this is how model selection criteria play a role as all model selection criteria such as the Akaike Information Criterion and the Bayesian Information Criterion are based to some extent on the principle of parsimony (a balance between the model accuracy and the estimation accuracy).…”

“…Actually, this is how model selection criteria play a role as all model selection criteria such as the Akaike Information Criterion and the Bayesian Information Criterion are based to some extent on the principle of parsimony (a balance between the model accuracy and the estimation accuracy). We again refer the interested reader to [7,19] for more information on this. Using a model selection criteria to select a best value for N is an area for current investigations [7].…”

“…We again refer the interested reader to [7,19] for more information on this. Using a model selection criteria to select a best value for N is an area for current investigations [7].…”

“…Hence, based on the above discussion we know that if both K and Θ are assumed to be compact, then there exists a solution to (3.4) or (3.5). Under usual standard assumptions on h and how data is sampled, the consistency of estimators can also be established (see our companion theoretical paper [7] for details).…”

Method Comparison for Estimation of Distributed Parameters in Permittivity Models Using Reflectance

“…In a case where the material under study is inorganic glass, a convolution of the Lorentz and Gaussian functions (a linear combination of normal distributions is imposed on the resonance frequency in the Lorentz model) was proposed by Efimov, et al, as early as in 1985 (e.g., see [12,13]) (we will refer to this as the Efimov approach). Another possible approach to deal with this difficulty, which was investigated in [4,5], is to impose an unknown probability distribution on the dielectric parameters. In that work, a distribution was imposed on the resonance wavenumber and we continue that convention in our current investigation.…”