Nonparametric empirical likelihood density functionals estimation and applications

Abstract

Chapter 2 of this dissertation presents a nonparametric empirical likelihood estimation of kernel density functionals (ELKDFE), which are constructed based on a kernel density functional estimation (KDFE) and the concepts of empirical likelihood. The work focuses on estimating the integration of square density function and a known function which has a derivative of order p, for p > 0. In many applications there may be extra information available to use, hence the concept of empirical likelihood becomes useful in providing a systematic approach for capturing the extra information. So ELKDFE reduces the MSE, especially when the sample size is small to moderate, and the difference of MSE between those two estimates decreases as the sample size increases.

Secondly, in Chapters 3 and 4, two new kernel estimators are proposed, GCA and LCA, and their rationales, properties, empirical likelihood versions, data-driven bandwidth selection, and applications are given as well. The bandwidth of the new approach is much tighter, catching the density's humps and valleys is more accurate. These estimates can be used for fixed and sequential sampling. The empirical likelihood (EL) versions of the GCA and LCA are provided and shown to have smaller AMISE than that of the non-EL estimation, and the difference of MISE tends to shrink as the sample size increases.

The GCA and LGA estimates are applied to regression using a local polynomial setting. It is shown that the regression estimators based on GCA and LGA have smaller bias and variance than standard kernel regression estimators.

An investigation of the properties of cumulative distribution function estimation based on GCA and LGA shows that the new estimators have smaller MSE and better performance than standard kernel CDF estimation.