Examples and Problems in Mathematical Statistics

CHAPTER 3

Sufficient Statistics and the Information in Samples

PART I: THEORY

3.1 INTRODUCTION

The problem of statistical inference is to draw conclusions from the observed sample on some characteristics of interest of the parent distribution of the random variables under consideration. For this purpose we formulate a model that presents our assumptions about the family of distributions to which the parent distribution belongs. For example, in an inventory management problem one of the important variables is the number of units of a certain item demanded every period by the customer. This is a random variable with an unknown distribution. We may be ready to assume that the distribution of the demand variable is Negative Binomial NB(, ν). The statistical model specifies the possible range of the parameters, called the parameter space, and the corresponding family of distributions . In this example of an inventory system, the model may be

Such a model represents the case where the two parameters, and ν, are unknown. The parameter space here is Θ = {(, ν); 0 < < 1, 0 < ν < ∞ }. Given ...