# Elements of Hypothesis Testing

## Abstract

Most signal detection problems can be cast in the framework of *M*-*ary hypothesis testing*,in which we have an observation (possibly a vector or function) on the basis of which we wish to decide among *M* possible statistical situations describing the observations. For example, in an *M*-ary communications receiver we observe an electrical waveform that consists of one of *M* possible signals corrupted by random channel or receiver noise, and we wish to decide which of the *M* possible signals is present. Obviously, for any given decision problem, there are a number of possible decision strategies or rules that could be applied; however, we would like to choose a decision rule that is optimum in some sense. There are several useful definitions of optimality for such problems, and in this chapter we consider the three most common formulations—Bayes, minimax, and Neyman-Pearson—and derive the corresponding optimum solutions. In general, we consider the particular problem of binary (*M* = 2) hypothesis testing, although the extension of many of the results of this chapter to the general *M*-ary case is straightforward and will be developed in the exercises. The application of this theory to those models specific to signal detection is considered in detail in Chapters III and VI.

## Keywords

Decision Rule Minimax Risk Uniformly Much Powerful Bayesian Hypothesis Uniform Cost## Preview

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