An introduction to Bartlett correction and bias reduction
This monograph deals with two important topics of research among statisticians. The first one is the Bartlett and Bartlett-type corrections that can be applied to test statistics in order to bring the empirical sizes of asymptotic tests close to the corresponding nominal sizes, while the second one is the strategies that can be employed to reduce the bias of the maximum likelihood estimator. The underlying idea behind both groups of corrections is to obtain higher accuracy in small samples. Even though the focus of the authors is to present, in a coherent way, analytical corrections to test statistics and estimators, they also discuss and provide numerical alternatives, which are mainly based on bootstrap resampling, that can be used for the same purpose. In this statistical framework, the monograph consists of four chapters, an appendix, where supplementary material is given, and a glossary. In the sequel, the subject of each one of the four chapters is briefly presented. Chapter 1 (Likelihood-based inference and finite-sample corrections: a brief overview) briefly introduces the likelihood function, maximum likelihood estimators and some of their properties as well as the most commonly asymptotic testing criteria, namely the likelihood ratio (LR), the score test and the Wald test. Moreover, some remarks related to concepts that will be used throughout the book are given. Chapter 2 (Bartlett corrections and bootstrap testing inference) initially provides a unified review of the literature on Bartlett correction to the likelihood ratio test statistic. Afterwards, a discussion on how to obtain the corrections in regression models, such as generalized linear models (GLM) and Birnbaum-Saunders nonlinear regression models is given. Finally, it is described how the bootstrap resampling scheme can be used as an alternative to analytical finite sample corrections for improving on LR testing inference. Chapter 3 (Bartlett-type corrections) provides a unified review of the literature on Bartlett-type corrections to chi-squared asymptotic criteria other than the likelihood ratio statistic. In this frame, the authors restrict their presentation to the score test and the Wald test, since these two tests and the LR are the most commonly used large sample testing criteria. An application to generalized linear models and heteroskedastic linear regression is presented. Finally, using the linear regression framework, the authors provide simulation studies that show how the independent variables and the number of nuisance parameters can affect the first-order asymptotic approximation to some test statistics. Chapter 4 (Analytical and bootstrap bias corrections) initially provides two analytical approaches that can be used to reduce the bias of the maximum likelihood estimator. Afterwards, numerical strategies for bias-correcting parameter estimators based on bootstrap are discussed. Bias corrections in different statistical models are presented and discussed. Summarizing, this monograph endeavors to give a review of research on the topic of Bartlett and Bartlett-type corrections that can be applied to test statistics as well as bias corrections of maximum likelihood estimators. The authors have written an interesting book, which is intended to serve the need of researchers and graduate students in statistics. The book could also be very useful as a supplement for graduate level courses among others in statistical inference. At the end of each chapter, references are provided and this will help a reader wishing to pursue this area further, since details involved in many of the derivations are not included. The material of the monograph is mainly based on a series of publications written by the authors and their colleagues-coauthors.
scientific article; zbMATH DE number 6305058