Question

Let \(X_1,X_2\) be a random sample of size \(2\) from an \(Exp(\theta)\) distribution, where \(\theta>0\) is an unknown parameter. Let

• A. \(T_1\) is an unbiased estimator of \(e^{-2/\theta}\)
• B. \(T_2\) is the uniformly minimum variance unbiased estimator of \(e^{-2/\theta}\)
• C. \(Var(T_1)\leq \dfrac{2}{\theta}e^{-4/\theta}\) for all \(\theta>0\)
• D. \((X_1,X_2)\) is a complete sufficient statistic

Hint:

To find a UMVUE, first find an unbiased estimator and then take its conditional expectation given a complete sufficient statistic.

Answer 1

The Correct Option is A, B

Step 1: Unbiasedness of $T_1$ (A).
With $T_1=\mathbf 1(X_1>2)$, $E(T_1)=P(X_1>2)=e^{-2/\theta}$, so $T_1$ is unbiased for $e^{-2/\theta}$. (A) holds.

Step 2: Complete sufficient statistic.
$S=X_1+X_2\sim Gamma(2,\theta)$ is complete and sufficient.

Step 3: Rao-Blackwell for $T_2$ (B).
Given $S=s$, $X_1$ is uniform on $(0,s)$, so $E(T_1|S=s)=P(X_1>2|S=s)=\max\{0,\frac{s-2}{s}\}$. This is exactly $T_2$, so $T_2$ is the UMVUE. (B) holds.

Step 4: Knock out (C) and (D).
$\mathrm{Var}(T_1)=e^{-2/\theta}(1-e^{-2/\theta})$ is not bounded by $\frac2\theta e^{-4/\theta}$ for all $\theta$, so (C) fails. And $(X_1,X_2)$ is sufficient but not complete (since $E(X_1-X_2)=0$ without $X_1-X_2$ being $0$), so (D) fails.

Step 5: Collect.
\[ \boxed{(A),(B)} \]