Examples and Problems in Mathematical Statistics

CHAPTER 5

Statistical Estimation

PART I: THEORY

5.1 GENERAL DISCUSSION

Point estimators are sample statistics that are designed to yield numerical estimates of certain characteristics of interest of the parent distribution. While in testing hypotheses we are generally interested in drawing general conclusions about the characteristics of the distribution, for example, whether its expected value (mean) is positive or negative, in problems of estimation we are concerned with the actual value of the characteristic. Generally, we can formulate, as in testing of hypotheses, a statistical model that expresses the available information concerning the type of distribution under consideration. In this connection, we distinguish between parametric and nonparametric (or distribution free) models. Parametric models specify parametric families of distributions. It is assumed in these cases that the observations in the sample are generated from a parent distribution that belongs to the prescribed family. The estimators that are applied in parametric models depend in their structure and properties on the specific parametric family under consideration. On the other hand, if we do not wish, for various reasons, to subject the estimation procedure to strong assumptions concerning the family to which the parent distribution belongs, a distribution free procedure may be more reasonable. In Example 5.1, we illustrate some of these ideas.

This chapter is devoted to the theory and applications of these ...