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# ISI MStat PSB 2012 Problem 10 | MVUE Revisited

This is a very simple sample problem from ISI MStat PSB 2012 Problem 10. It's a very basic problem but very important and regular problem for statistics students, using one of the most beautiful theorem in Point Estimation. Try it!

## Problem- ISI MStat PSB 2012 Problem 10

Let $X_1,X_2,.....X_{10}$ be i.i.d. Poisson random variables with unknown parameter $\lambda >0$. Find the minimum variance unbiased estimator of exp{$-2\lambda$}.

### Prerequisites

Poisson Distribution

Minimum Variance Unbiased Estimators

Lehman-Scheffe's Theorem

Completeness and Sufficiency

## Solution :

Well, this is a very straight forward problem, where we just need to verify certain conditions, of sufficiency and completeness.

If, one is aware of the nature of Poisson Distribution, one knows that for a given sample $X_1,X_2,.....X_{10}$, the sufficient statistics for the unknown parameter $\lambda>0$, is $\sum_{i=1}^{10} X_i$ , also by extension $\sum_{i}X_i$ is also complete for $\lambda$ (How??).

So, now first let us construct an unbiased estimator of $e^{-2\lambda}$. Here, we need to observe patterns as usual. Let us define an Indicator Random variable,

$I_X(x) = \begin{cases} 1 & X_1=0\ and\ X_2=0 \\ 0 & Otherwise \end{cases}$,

So, $E(I_X(x))=P(X_1=0, X_2=0)=e^{-2\lambda}$, hence $I_X(x)$ is an unbiased estimator of $e^{-2\lambda}$. But is it a Minimum Variance ??

Well, Lehman-Scheffe answers that, Since we know that $\sum X_i$ is complete and sufficient for $\lambda$, By Lehman-Scheffe's theorem,

$E(I_X(x)|\sum X_i=t)$ is the minimum variance unbiased estimator of $e^{-2\lambda }$ for any $t>0$. So, we need to find the following,

$E(I_X(x)|\sum_{i=1}^{10}X_i=t)= \frac{P(X_1=0,X_2; \sum_{i}X_i=t)}{P(\sum_{i=3}^{10}X_i=t)}=\frac{e^{-2\lambda}e^{-8\lambda}\frac{(8\lambda)^t}{t!}}{e^{10\lambda}\frac{(10\lambda)^t}{t!}}=(\frac{8}{10})^t$.

So, the Minimum Variance Unbiased Estimator of exp{$-2\lambda$} is $(\frac{8}{10})^{\sum_{i=1}^{10}X_i}$

Now can you generalize this for a sample of size n, again what if I defined $I_X(x)$ as,

$I_X(x) = \begin{cases} 1 & X_i=0\ &\ X_j=0 \\ 0 & Otherwise \end{cases}$, for some $i \neq j$,

would it affected the end result ?? What do you think?

## Food For Thought

Let's not end our concern for Poisson, and think further, that for the given sample if the sample mean is $\bar{X}$ and sample variance is $S^2$. Can you show that $E(S^2|\bar{X})=\bar{X}$, and further can you extend your deductions to $Var(S^2) > Var(\bar{X})$ ??

Finally can you generalize the above result ?? Give some thoughts to deepen your insights on MVUE.

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