Tests of Hypotheses for Model Parameters
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This chapter considers the problem of hypothesis testing. Starting from first principles, it develops the Neyman-Pearson framework, and discusses different forms of hypothesis pairs. It then uses the question of optimality of tests as a means for motivating specific test functions. This is done in a general setup for simple-vs-simple hypotheses, and in an exponential family context for one-sided alternatives. The likelihood ratio method and Wald’s method are then introduced, to address the more general case of two-sided alternatives. Asymptotic approximations are then used in order to determine approximate critical values in the context of exponential families. The chapter concludes with the introduction of p-values, their basic properties, and their relation to the Neyman-Pearson paradigm.