Chapter 4 Hypothesis Testing

(I wasn’t entirely happy with the treatment of Wasserman and Casella & Berger. So I’m following the treatment of Hogg, McKean & Craig.)

A hypothesis test is a process to reject or not reject a well-defined statements. Intuitively, there are three components to a hypothesis test:

  1. Null hypothesis \(H_0\)versus Alternative hypothesis \(H_1\)
  2. Data
  3. Decision rule to reject \(H_0\) and accept \(H_1\) or to not reject \(H_0\) and reject \(H_1\).

The mathematical formulation of this is a bit more restrictive because of the need for well-defined and verifiable statements. We will restrict our attention to hypotheses about either:

  1. a parameter of a model, or
  2. a functional of the underlying density distribution \(f\), i.e., a mapping \(T(f) \in \mathbb{R}\). For example, \(\mu(f) = \int x f(x) dx\).