Pseudo-Maximization in Likelihood and Bayesian Inference

The self-normalized statistics in Chaps. 15 and 16 are Studentized statistics of the form $(\hat \theta - \theta )/\hat {se}$ which are generalizations of the t-statistic for ${\sqrt n} {(\bar X_n - \mu )}{/s_n}$ testing the null hypothesis that the mean of a normal distribution is μ, when the variance ō² is unknown and estimated by the sample variance s²_n. In Sect. 17.1 we consider another class of self-normalized statistics, called generalized likelihood ratio (GLR) statistics, which are extensions of likelihood ratio (LR) statistics (for testing simple hypotheses) to composite hypotheses in parametric models. Whereas LR statistics are martingales under the null hypothesis, GLR statistics are no longer martingales but can be analyzed by using LR martingales and the pseudo-maximization technique of Chap. 11. The probabilistic technique of pseudo-maximization via the method of mixtures has a fundamental statistical counterpart that links likelihood to Bayesian inference; this is treated in Sect. 17.2.