Statistical Inference for Models with Multivariate t-Distributed Errors

A. K. Md. Ehsanes Saleh, Mohammad Arashi, S M M Tabatabaey

• Publisher: Wiley
• Year: 2014
• Print ISBN: 9781118854051
• eBook ISBN: 9781118853962
• Edition: 1
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

This book summarizes the results of various models under normal theory with a brief review of the literature. Statistical Inference for Models with Multivariate t-Distributed Errors:

• Includes a wide array of applications for the analysis of multivariate observations
• Emphasizes the development of linear statistical models with applications to engineering, the physical sciences, and mathematics
• Contains an up-to-date bibliography featuring the latest trends and advances in the field to provide a collective source for research on the topic
• Addresses linear regression models with non-normal errors with practical real-world examples
• Uniquely addresses regression models in Student's t -distributed errors and t -models
• Supplemented with an Instructor's Solutions Manual, which is available via written request by the Publisher

CHAPTER 1

INTRODUCTION

Outline

1.1 Objective of the Book

1.2 Models under Consideration

1.3 Organization of the Book

1.4 Problems

1.1 Objective of the Book

The classical theory of statistical analysis is primarily based on the assumption that the errors of various models are normally distributed. The normal distribution is also the basis of the (i) chi-square, (ii) Student’s t-, and (iii) F-distributions. Fisher (1956) pointed out that slight differences in the specification of the distribution of the model errors may play havoc on the resulting inferences. To examine the effects on inference, Fisher (1960) analyzed Darwin’s data under normal theory and later under a symmetric non-normal distribution. Many researchers have since investigated the influence on inference of distributional assumptions differing from normality. Further, it has been observed that most economic and business data, e.g., stock return data, exhibit long-tailed distributions. Accordingly, Fraser and Fick (1975) analyzed Darwin’s data and Baltberg and Gonedes (1974) analyzed stock returns using a family of Student’s t-distribution to record the effect of distributional assumptions compared to the normal theory analysis. Soon after, Zellner (1976) considered analyzing stock return data by a simple regression model, assuming the error distribution to have a multivariate t-distribution. He revealed the fact that dependent but uncorrelated responses can be analyzed by multivariate t-distribution. He discussed differences as well as similarities of the results in both classical and Bayesian contexts for multivariate normal and multivariate t-based models.

Fraser (1979, p. 37) emphasized that the normal distribution is extremely short-tailed and thus unrealistic as a sole distribution for variability. He demonstrated the robustness of the Student’s t-family as opposed to the normal distribution based on numerical studies. In justifying the appropriateness and the essence of the use of Student’s t-distribution, Prucha and Kal...

Table of contents

• Cover
• Half Title page
• Title page
• Copyright page
• Dedication
• List of Figures
• List of Tables
• Preface
• Glossary
• List of Symbols
• Chapter 1: Introduction
• Chapter 2: Preliminaries
• Chapter 3: Location Model
• Chapter 4: Simple Regression Model
• Chapter 5: Anova
• Chapter 6: Parallelism Model
• Chapter 7: Multiple Regression Model
• Chapter 8: Ridge Regression
• Chapter 9: Multivariate Models
• Chapter 10: Bayesian Analysis
• Chapter 11: Linear Prediction Models
• Chapter 12: Stein Estimation
• References
• Author Index
• Subject Index