Question

Let \(X\) be a random sample of size \(1\) from a \(U(0,\theta)\) distribution, where \(\theta>0\) is an unknown parameter. To test \(H_0:\theta\geq 1\) against \(H_1:\theta<1\), the critical region \(X<\frac{3}{4}\) is being used. If \(\beta(\cdot)\) is the power function of the test, then which one of the following statements is NOT true?

• A. The size of the test is \(\dfrac{3}{4}\)
• B. \(\inf_{\theta<1}\beta(\theta)=\dfrac{3}{4}\)
• C. For all \(\theta>0,\;\beta(\theta)<1\)
• D. \(\displaystyle \lim_{\theta\to 0}\beta(\theta)=1\)

Hint:

For a \(U(0,\theta)\) distribution, probabilities are obtained by dividing interval length by \(\theta\). Always write the power function piecewise depending on the relation between the cutoff and \(\theta\).

Answer 1

The Correct Option is C

Step 1: Write the power function.
With $X\sim U(0,\theta)$ and rejection $X<\tfrac34$, $\beta(\theta)=P_\theta(X<\tfrac34)$, which is $1$ for $\theta\le\tfrac34$ and $\tfrac{3}{4\theta}$ for $\theta>\tfrac34$.

Step 2: Check the size (A).
The size is $\sup_{\theta\ge1}\beta(\theta)$. Since $\tfrac{3}{4\theta}$ falls as $\theta$ grows, the max is at $\theta=1$, giving $\tfrac34$. So (A) is true.

Step 3: Check (B).
For $\theta<1$ the smallest power is the limit as $\theta\to1^-$, namely $\tfrac34$. So (B) is true.

Step 4: Check (C) and (D).
For $\theta\le\tfrac34$, $\beta(\theta)=1$, so 'always less than $1$' is false: (C) is the untrue one. As $\theta\to0$, $\beta=1$, so (D) is true.

Step 5: Conclude.
\[ \boxed{(C)} \]