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Mathematical Statistics and Stochastic Processes by Denis Bosq

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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 images and a partition of Θ into two subsets Θ0 and Θ1, we must decide, in light of an observation xE, whether the true value of θ is in Θ0 or in Θ1.

images is called the null hypothesis and images 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 images, or indeed to test images against images. 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 ...

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