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Chapter 9: Q18E (page 530)

Consider the situation described immediately before Eq. (9.1.12). Prove that the expression (9.1.12) equals the smallest伪0such that we would reject H0 at level of significance伪0.

Short Answer

Expert verified

\({P_\theta }(T \ge t)\)is non-increasing in \(t\), thus, the smallest \({\alpha _0}\)that rejects the null hypothesis is \(\mathop {\sup }\limits_{\theta \in {\Omega _0}} {P_\theta }(T \ge t)\).

Step by step solution

01

Definition of the p-value

A p-value is defined as the smallest 伪0 such that the null hypothesis is rejected at level 伪0 with the observed data.

02

Proving the function

The test 未t rejects the null for large values of a statistics T. Now, P胃( T鈮) is non-increasing in t for any statistic T. Thus, the smallest probability under the null hypothesis that未t rejects the null hypothesis occurs at

\begin{aligned}Sup\underset{_{\theta e}\Omega _{0}}{P_{\theta}}(T\geq t)\end{aligned}

This quantity equals its p-value by its very definition, so our argument is complete.

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In Exercise 8, assume that Z=z is observed. Find a formula for thep-value.

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