Hypothesis-Testing Methods

Abstract

In this chapter we examine general methods that can be used to define explicit rules for testing statistical hypotheses. In particular, the likelihood ratio, Wald, and Lagrange multiplier methods for constructing statistical tests are widely used in empirical work, and they provide well-defined procedures for defining test statistics and critical regions in given hypothesis- testing contexts. In addition, it is possible to find useful test statistics based entirely on a heuristic principle of test construction. None of these four methods is guaranteed to produce a statistical test with optimal properties in all cases. In fact, no method of defining statistical tests can provide such a guarantee. The virtues of these methods are that they are relatively straightforward to apply (in comparison to direct implementation of many of the theorems in Section 9.5), they are applicable to a wide class of problems that are relevant in applications, they generally have excellent asymptotic properties, they often have good power in finite samples, they are sometimes unbiased and/or UMP, and they have intuitive appeal.