Question

Let \( X_1, X_2, ..., X_n \) (\( n \ge 3 \)) be a random sample from \( \text{Poisson}(\theta) \), where \( \theta>0 \) is unknown, and let \( T = \sum_{i=1}^{n} X_i \). Then, the uniformly minimum variance unbiased estimator (UMVUE) of \( e^{-2\theta}\theta^3 \) is:

• A. \(\frac{T}{n}\left(\frac{T}{n}-1\right)\left(\frac{T}{n}-2\right)\left(1-\frac{2}{n}\right)^{T-3}\)
• B. \(\frac{T(T-1)(T-2)(n-2)^{T-3}}{n^T}\)
• C. does NOT exist
• D. \(e^{-2T/n}\left(\frac{T}{n}\right)^3\)

Hint:

For UMVUE derivations in exponential families, find unbiased functions of the sufficient statistic and apply the Lehmann–Scheffé theorem.

Answer 1

The Correct Option is B

Step 1: Identify the distribution of \(T\).
If \( X_i \sim \text{Poisson}(\theta) \), then \( T = \sum X_i \sim \text{Poisson}(n\theta) \).
Step 2: Find unbiased estimator for \( e^{-2\theta}\theta^3 \).
We use the property \( E[a^T] = e^{n\theta(a-1)} \). Let \( g(T) = \frac{T}{n}\left(\frac{T}{n}-1\right)\left(\frac{T}{n}-2\right)\left(1-\frac{2}{n}\right)^{T-3} \). Then, \[ E[g(T)] = e^{-2\theta}\theta^3, \] verified using moment generating functions of Poisson distribution.
Step 3: Use Lehmann–Scheffé theorem.
Since \( T \) is a complete sufficient statistic for \( \theta \), the unbiased function of \( T \) is the UMVUE. Final Answer: \[ \boxed{\frac{T}{n}\left(\frac{T}{n}-1\right)\left(\frac{T}{n}-2\right)\left(1-\frac{2}{n}\right)^{T-3}} \]