Quadratic Mean Differentiable Families

As mentioned at the beginning of Chapter 11, the finite sample theory of optimality for hypothesis testing applied only to rather special parametric families, primarily exponential families and group families. On the other hand, asymptotic optimality will apply more generally to parametric families satisfying smoothness conditions. In particular, we shall assume a certain type of differentiability condition, called quadratic mean differentiability. Such families will be considered in Section 12.2. In Section 12.3, the notion of contiguity will be developed, primarily as a technique for calculating the limiting distribution or power of a test statistic under an alternative sequence, especially when the limiting distribution under the null hypothesis is easy to obtain. In Section 12.4, these techniques will then be applied to classes of tests based on the likelihood function, namely the Wald, Rao, and likelihood ratio tests. The asymptotic optimality of these tests will be established in Chapter 13.