An Introduction to Probability and Statistics, 3rd Edition

7BASIC ASYMPTOTICS: LARGE SAMPLE THEORY

7.1 INTRODUCTION

In Chapter 6 we described some methods of finding exact distributions of sample statistics and their moments. While these methods are used in some cases such as sampling from a normal population when the sample statistic of interest is or S², often either the statistics of interest, say , is either too complicated or its exact distribution is not simple to work with. In such cases we are interested in the convergence properties of T_n. We want to know what happens when the sample size is large. What is the limiting distribution of T_n? When the exact distribution of T_n (and its moments) is unknown or too complicated we will often use their asymptotic approximations when n is large.

In this chapter, we discuss some basic elements of statistical asymptotics. In Section 7.2 we discuss various modes of convergence of a sequence of random variables. In Sections 7.3 and 7.4 the laws of large numbers are discussed. Section 7.5 deals with limiting moment generating functions and in Section 7.6 we discuss one of the most fundamental theorem of classical statistics called the central limit theorem. In Section 7.7 we consider some statistical applications of these methods.

The reader may find some parts of this chapter a bit difficult ...