Statistical Inference for Models with Multivariate t-Distributed Errors

In problems of statistical inference, it is customary to use normal distribution as the basis of statistical analysis. Many results related to univariate analysis can be extended to multivariate analysis using multidimensional normal distribution, and statisticians have tried to broaden the scope of the distributions and achieve reasonable inferential conclusions.  Zellner (1976) introduced the idea of using Student’s t-distribution, which can accommodate the heavier tailed distributions in a reasonable way and producing robust inference procedures for applications. Most of the research with Student’s t-distribution, so far, is focused on the agreement of the results with that of the normal theory. For example, the MLE of the location parameter agrees with of the mean-vector of a normal distribution. Similarly, the likelihood ratio test under the Student’s t-distribution has same distribution as the normal distribution under the null hypothesis.  This book consists of thirteen chapters. Chapter 1 summarizes the results of various models under normal theory with brief review of the literature. Chapter 2 contains the basic properties of various known distributions and opens discussion of multivariate t-distribution and elliptically contoured distributions with their basic properties. Chapter 3 discusses the statistical analysis of a location model from estimation of the intercept and slope to test of hypothesis of the parameters. The authors also add the preliminary test and shrinkage type estimators of the three parameters, which include the estimation of the scale parameter of the model while Chapter 4 contains similar details of a simple regression model. Chapter 5 is devoted to ANOVA models and discussing on preliminary test and shrinkage type estimators in elliptically contoured distributions, and Chapter 6 deals with the parallelism model in the same spirit.  Multiple regression models are discussed in Chapter 7, and ridge regression is addressed in Chapter 8.  Statistical inference of multivariate models and simple multivariate linear models are discussed in Chapter 9 and 10. Bayesian analysis is discussed in elliptically contoured models in Chapter 11, and the statistical analysis of linear prediction models is included in chapter 12. The book concludes with Chapter 13, which is devoted to shrinkage estimation.  Additional topical coverage includes: location models; simple regression models; ANOVA; paralllelism models; multiple regression models; ridge regression; multivariate models; simple multivariate linear models; Bayesian analysis; linear prediction models; and Stein estimation.