Скачать или смотреть Finding the MVU Estimator for a Function of Poisson Parameter | UPSC ISS 2024 Paper-2 | Problem-16

This video presents the derivation of a minimum variance unbiased (MVU) estimator for a function of the Poisson parameter \(\theta\) by leveraging the complete and sufficient statistic \(T = \sum_{i=1}^n X_i\), where \(X_i\) are independent Poisson(\(\theta\)) random variables. Key steps include:

Recognizing \(T \sim \text{Poisson}(n\theta)\), a complete and sufficient statistic for \(\theta\).
Using the Lehmann-Scheffé theorem to assert that any unbiased estimator based on \(T\) is MVU.
Formulating the expectation condition
\[
E[h(T)] = e^{-2\theta} \theta^3,
\]
and solving it through expansion and coefficient matching.
Deriving the explicit form of \(h(T)\) for \(T \geq 3\), which involves factorial and power terms.

The final MVU estimator is expressed as
\[
h(T) = \frac{T(T-1)(T-2)(n-2)^{T-3}}{n^T},
\]
corresponding to option D. This example illustrates how to construct unbiased estimators of nonlinear functions in Poisson models with efficient variance properties.

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