Revival: Large Sample Methods in Statistics (1994)

An Introduction with Applications

Pranab K. Sen, Julio M. Singer

• Publisher: CRC Press
• Year: 2017
• Print ISBN: 9781138106017
• eBook ISBN: 9781351361163
• Edition: 1
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

This text bridges the gap between sound theoretcial developments and practical, fruitful methodology by providing solid justification for standard symptotic statistical methods. It contains a unified survey of standard large sample theory and provides access to more complex statistical models that arise in diverse practical applications.

CHAPTER 1

Objectives and Scope: General
Introduction

1.1 Introduction

Large sample methods in Statistics constitute the general methodology underlying fruitful simpler statistical analyses of data sets involving a large number of observations. Drawing statistical conclusions from a given data set involves the choice of suitable statistical models relating to the observations which incorporate some random (stochastic) or chance factors whereby convenient probability laws can be adopted in an appropriate manner. It is with respect to such postulated probability laws that the behavior of some sample statistics (typically, an estimator in an estimation problem or a test statistic in a hypothesis testing problem) needs to be studied carefully so that the conclusions can be drawn with an adequate degree of precision. If the number of observations is small and/or the underlying probability model is well specified, such stochastic behavior can be evaluated in an exact manner. However, with the exception of some simple underlying probability laws (such as the normal or Poisson distributions), the exact sampling distribution of a statistic may become very complicated as the number of observations in the sample becomes large. Moreover, if the data set actually involves a large number of observations, there may be a lesser need to restrict oneself to a particular probability law, and general statistical conclusions as well can be derived by allowing such a law to be a member of a broader class. In other words, one may achieve more robustness with respect to the underlying probability models when the number of observations is large. On the other hand, there are some natural (and minimal) requirements for a statistical method to qualify as a valid large sample method. For example, in the case of an estimator of a parameter, it is quite natural to expect that as the sample size increases, the estimator should be closer to the parameter in some meaningful sense; in the literature, this property is known as the consistency of estimators. Similarly, in testing a null hypothesis, a test should be able...

Table of contents

• Cover Page
• Half Title Page
• Title Page
• Copyright
• Dedication
• Table of Contents
• Preface
• 1 Objectives and Scope: General Introduction
• 2 Stochastic Convergence
• 3 Weak Convergence and Central Limit Theorems
• 4 Large Sample Behavior of Empirical Distributions and Order Statistics
• 5 Asymptotic Behavior of Estimators and Test Statistics
• 6 Large Sample Theory for Categorical Data Models
• 7 Large Sample Theory for Regression Models
• 8 Invariance Principles in Large Sample Theory
• References
• Index