# 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.

## Keywords

Decision Rule Conditional Risk Hypothesis Testing Problem Minimax Risk Binary Channel## Preview

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