Bartlett-Type Adjustments, Comparisons of Tests, and Recent Developments of Asymmetric Kernel Density Estimation

Abstract

This paper deals with two topics. One is higher-order local power comparison of tests related to Bartlett-type adjustments, and the other is nonparametric kernel type density estimation. It is well known that the chi-squared approximation for the log likelihood ratio (LR) statistic is improved via mean correction, i.e., division (equivalently, multiplication) of the LR statistic by a suitable constant. We begin with the Bartlett correctability of the LR statistic and introduce general methods of improving the chi-squared approximations for several famous statistics rather than the LR statistic. We then briefly discuss comparisons of tests under a sequence of local alternatives. On the other hand, we also review recent issues on asymmetric kernel density estimation for nonnegative data, which is a remedy of the boundary bias problem of the commonly used location-scale-type Rosenblatt–Parzen kernel method.