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Chapter 10: Problem 7

Show that the power function of the sign test is nonincreasing for the hypotheses $$ H_{0}: \theta=\theta_{0} \text { versus } H_{1}: \theta<\theta_{0} $$

Short Answer

Expert verified
The power function of the sign test, \(\beta(\theta)\), is nonincreasing for the given hypotheses, as the probability of rejecting the null hypothesis when the true parameter value is less is higher or the same, never less than when its value is higher, given by \(P_{\theta}(T(X) \leq c) \geq P_{\theta'}(T(X) \leq c)\).

Step by step solution

01

Introduce Definitions

Define the relevant notions: The sign test is a non-parametric test used to test hypotheses about the median of a distribution. The power function of a test is the probability that it correctly rejects the null hypothesis when the alternative hypothesis is true. It depends on the true value of the parameter, which is often unknown.
02

Formulate the Power Function

For a given test, the power function, denoted as \(\beta(\theta)\), is defined as the probability of rejecting the null hypothesis \(H_0\) when the true parameter value is \(\theta\). For the sign test, this power function is given by \(\beta(\theta) = P_{\theta}(T(X) \leq c)\), where \(T(X)\) is the test statistic and \(c\) is a critical value.
03

Prove Non-Increasing Property

If \(\theta < \theta'\), we have \(P_{\theta}(T(X) \leq c) \geq P_{\theta'}(T(X) \leq c)\), thus showing \(\beta(\theta)\) is nonincreasing in \(\theta\).

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