An Introduction to Probability and Statistics, 3rd Edition

9NEYMAN–PEARSON THEORY OF TESTING OF HYPOTHESES

9.1 INTRODUCTION

Let X₁, X₂, …, X_n be a random sample from a population distribution F_θ, , where the functional form of F_θ is known except, perhaps, for the parameter θ. Thus, for example, the X_i’s may be a random sample from (θ,1), where is not known. In many practical problems the experimenter is interested in testing the validity of an assertion about the unknown parameter θ. For example, in a coin-tossing experiment it is of interest to test, in some sense, whether the (unknown) probability of heads p equals a given number , Similarly, it is of interest to check the claim of a car manufacturer about the average mileage per gallon of gasoline achieved by a particular model. A problem of this type is usually referred to as a problem of testing of hypotheses and is the subject of discussion in this chapter. We will develop the fundamentals of Neyman–Pearson theory. In Section 9.2 we introduce the various concepts involved. In Section 9.3 the fundamental Neyman–Pearson lemma is proved, and Sections 9.4 and 9.5 deal with some basic ...