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.