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: Check option (A).
For \(X_1\sim Exp(\theta)\), where \(\theta\) is the scale parameter,
\[ P(X_1>x)=e^{-x/\theta} \] Therefore,
\[ E(T_1)=P(X_1>2)=e^{-2/\theta} \] Hence, \(T_1\) is an unbiased estimator of \(e^{-2/\theta}\).
So, option (A) is true.

Step 2: Find a complete sufficient statistic.
For a random sample from \(Exp(\theta)\), the joint density depends on the sample through
\[ S=X_1+X_2 \] Thus,
\[ S=X_1+X_2 \] is sufficient for \(\theta\). Also, \(S\sim Gamma(2,\theta)\), and this gamma family is complete.
Hence, \(S\) is a complete sufficient statistic.

Step 3: Apply Rao-Blackwell theorem.
Since \(T_1\) is unbiased for \(e^{-2/\theta}\), the UMVUE is
\[ E(T_1\mid S) \] where
\[ S=X_1+X_2 \] Given \(S=s\), \(X_1\) is uniformly distributed on \((0,s)\).
Therefore,
\[ E(T_1\mid S=s)=P(X_1>2\mid S=s) \] If \(s\leq 2\), then
\[ P(X_1>2\mid S=s)=0 \] If \(s>2\), then
\[ P(X_1>2\mid S=s)=\frac{s-2}{s} \] Thus,
\[ E(T_1\mid S) = \max\left\{0,\frac{S-2}{S}\right\} \] Since \(S=X_1+X_2\),
\[ E(T_1\mid S) = \max\left\{0,\frac{X_1+X_2-2}{X_1+X_2}\right\}=T_2 \] Hence, \(T_2\) is the UMVUE of \(e^{-2/\theta}\).
So, option (B) is true.

Step 4: Check option (C).
Since \(T_1\) is an indicator random variable with
\[ P(T_1=1)=e^{-2/\theta}, \] we get
\[ Var(T_1)=e^{-2/\theta}\left(1-e^{-2/\theta}\right) \] This is not always less than or equal to
\[ \frac{2}{\theta}e^{-4/\theta} \] for all \(\theta>0\).
Hence, option (C) is false.

Step 5: Check option (D).
The statistic \((X_1,X_2)\) is sufficient, but it is not complete.
For example, a non-zero function such as
\[ h(X_1,X_2)=X_1-X_2 \] has expectation
\[ E(X_1-X_2)=0 \] for every \(\theta>0\), but \(h(X_1,X_2)\) is not almost surely zero.
Hence, \((X_1,X_2)\) is not complete.
So, option (D) is false.

Step 6: Final conclusion.
The true statements are
\[ \boxed{(A)\text{ and }(B)} \]