Testing Goodness of Fit

So far, the principal framework of this book has been optimality (either exact or asymptotic) in situations where both the hypothesis and the class of alternatives were specified by parametric models. In the present chapter, we shall take up the crucial problem of testing the validity of such models, the hypothesis of goodness of fit. For example, we would like to know whether a set of measurements X ₁,…,X _n is consonant with the assumption that the X’s are an i.i.d. sample from a normal distribution.

A difficulty in testing such a hypothesis is that the class of alternatives typically is enormously large and can no longer be described by a parametric model. As a result, although some asymptotic optimality results are presented, they are isolated; no general asymptotic optimality theory seems to exist for this problem. In fact, there is growing evidence, such as the results of Janssen (2000a) (see Theorem 14.6.2), that any test can achieve high asymptotic power against local or contiguous alternatives for at most a finite-dimensional parametric family.

Because of the importance of the problem of testing goodness of fit, we shall nevertheless consider this problem here. However, the focus will no longer be on optimality. Instead, we shall present some of the principal methods that have been proposed and study their relative strengths and weaknesses.