Question

Let \(X\) be a single observation from a distribution having probability density function
\[ f_\theta(x)=\begin{cases}\dfrac{2x}{\theta^2}&\text{if }0<x<\theta\\0&\text{otherwise,}\end{cases} \]
where \(\theta\in(0,\infty)\). For testing \(H_0:\theta\leq1\) against \(H_1:\theta>1\), at level of significance \(0.05\), let \(\beta_1\) be the size of the uniformly most powerful test and \(\beta_2\) be the power of the uniformly most powerful test at \(\theta=2\). Then \(10\beta_1+40\beta_2\) equals ______ (answer in integer).

Hint:

The UMP test for this monotone likelihood ratio family rejects for large \(X\); fix the cutoff using the boundary \(\theta=1\), then use it to find the power at \(\theta=2\).

Answer 1

Correct Answer: 31

Step 1: Transform T=X^2.
$T\sim$Uniform$(0,\theta^2)$.

Step 2: UMP in terms of T.
Reject when $T>c_0$.

Step 3: Fix c0.
At $\theta=1$: $T\sim$Uniform(0,1), $1-c_0=0.05\Rightarrow c_0=0.95$. $\beta_1=0.05$.

Step 4: Power at theta=2.
$T\sim$Uniform(0,4): $\beta_2=(4-0.95)/4=0.7625$.

Step 5: Combine.
\[ 10(0.05)+40(0.7625)=31 \]
\[ \boxed{31} \]