Discussion
Let \(X\) be a random sample of size \(1\) from a \(U(0, \theta)\) distribution, where \(\theta \gt 0\) is an unknown parameter. To test \(H_{0}: \theta \geq 1\) against \(H_{1}: \theta \lt 1\), the critical region \(X \lt \frac{3}{4}\) is being used. If \(\beta(\cdot)\) is the power function of the test, then which one of the following statements is NOT true?
Answer: C
The statement that is NOT true is for all \(\theta \gt 0\), \(\beta(\theta) \lt 1\).
The power function is \(\beta(\theta) = P_{\theta}\left(X \lt \frac{3}{4}\right)\):
\[ \beta(\theta) = \begin{cases} 1, & 0 \lt \theta \leq \frac{3}{4}, \\ \frac{3}{4\theta}, & \theta \gt \frac{3}{4}. \end{cases} \]
For \(\theta \leq \frac{3}{4}\) every observation is below \(\frac{3}{4}\), so the power equals 1 and the statement is false.
The other three are true. The size is \(\sup_{\theta \geq 1}\frac{3}{4\theta} = \frac{3}{4}\), attained at \(\theta = 1\). Over \(\theta \lt 1\), \(\beta\) decreases towards \(\frac{3}{4}\) as \(\theta \to 1\) without reaching it, so the infimum is \(\frac{3}{4}\). And \(\beta(\theta) = 1\) for all small \(\theta\), so the limit as \(\theta \to 0\) is 1.
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