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, and Craig 2019))

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 H0versus Alternative hypothesis H1
  2. Data
  3. Decision rule to reject H0 and accept H1 or to not reject H0 and reject H1.

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)R. For example, μ(f)=xf(x)dx.

References

Hogg, Robert V., Joseph W. McKean, and Allen T. Craig. 2019. Introduction to Mathematical Statistics. 8th ed. Boston: Pearson.