Chapter 7 - Limiting non-null distributions

Abstract

This chapter describes the different aspects of limiting non null distributions. The concept of contiguity of probability measures plays a basic role in the development of general asymptotic distribution of rank test statistics, under a suitable sequence of local alternatives. The concept of contiguity plays a fundamental role in the asymptotic theory of statistical inference. An intimately connected concept is the locally asymptotically quadratic (LAQ) family. The classical Hajek-LeCam-Inagaki convolution theorem is outlined. The asymptotic distribution of the likelihoods will regularly be log-normal. The asymptotic normality of log is defined in the same way as for an ordinary random variable. LeCam's second lemma remains applicable to single as well as multiple parameter models. It is found that in the case of a product measure, the observable random variables may also be vector valued, a case which arises in multivariate rank tests. It is observed that LeCam's third lemma plays a central role in the study of general asymptotic for rank statistics under contiguous alternatives.