Basic Large Sample Theory

Chapters 3-7 were concerned with the derivation of UMP, UMP unbiased, and UMP invariant tests. Unfortunately, the existence of such tests turned out to be restricted essentially to one-parameter families with monotone likelihood ratio, exponential families, and group families, respectively. Tests maximizing the minimum or average power over suitable classes of alternatives exist fairly generally, but are difficult to determine explicitly, and their derivation in Chapter 8 was confined primarily to situations in which invariance considerations apply.

Despite their limitations, these approaches have proved their value by application to large classes of important situations. On the other hand, they are unlikely to be applicable to complex new problems. What is needed for such cases is a simpler, less detailed, more generally applicable formulation. The development and implementation of such an approach will be the subject of the remaining chapters. It replaces optimality by asymptotic optimality obtained by embedding the actual situation in a sequence of situations of increasing sample size, and applying optimality to the limit situation. These limits tend to be of the simple type for which optimality has been established in earlier chapters.