Mathematical Statistics and Stochastic Processes

Chapter 6

Hypothesis Testing and Confidence Regions

6.1. Generalities

6.1.1. The problem

A hypothesis testing problem is defined in the following way: given a statistical model and a partition of Θ into two subsets Θ0 and Θ1, we must decide, in light of an observation x ∈ E, whether the true value of θ is in Θ0 or in Θ1.

is called the null hypothesis and is called the alternative. These two expressions demonstrate a dissymmetry in the way the problem is posed. In practice, we seek, in general, to verify the correctness of , or indeed to test against . This “dissymmetry” is accepted by most authors, and has given rise to the theory of hypothesis testing that Neyman and Pearson presented in a famous article in Biometrika (1928) [NEY 28].

6.1.2. Use of decision theory

In a hypothesis testing problem, the set of decisions may be written in the form D = {0,1}. A decision rule φ is called a ...