Get inspired by the success stories of our students in IIT JAM MS, ISI  MStat, CMI MSc DS.  Learn More

# 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 be i.i.d. Poisson random variables with unknown parameter . Find the minimum variance unbiased estimator of exp{ }.

### 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 , the sufficient statistics for the unknown parameter , is , also by extension is also complete for (How??).

So, now first let us construct an unbiased estimator of . Here, we need to observe patterns as usual. Let us define an Indicator Random variable, ,

So, , hence is an unbiased estimator of . But is it a Minimum Variance ??

Well, Lehman-Scheffe answers that, Since we know that is complete and sufficient for , By Lehman-Scheffe's theorem, is the minimum variance unbiased estimator of for any . So, we need to find the following, .

So, the Minimum Variance Unbiased Estimator of exp{ } is Now can you generalize this for a sample of size n, again what if I defined as, , for some ,

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 and sample variance is . Can you show that , and further can you extend your deductions to ??

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

## Subscribe to Cheenta at Youtube

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 be i.i.d. Poisson random variables with unknown parameter . Find the minimum variance unbiased estimator of exp{ }.

### 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 , the sufficient statistics for the unknown parameter , is , also by extension is also complete for (How??).

So, now first let us construct an unbiased estimator of . Here, we need to observe patterns as usual. Let us define an Indicator Random variable, ,

So, , hence is an unbiased estimator of . But is it a Minimum Variance ??

Well, Lehman-Scheffe answers that, Since we know that is complete and sufficient for , By Lehman-Scheffe's theorem, is the minimum variance unbiased estimator of for any . So, we need to find the following, .

So, the Minimum Variance Unbiased Estimator of exp{ } is Now can you generalize this for a sample of size n, again what if I defined as, , for some ,

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 and sample variance is . Can you show that , and further can you extend your deductions to ??

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

## Subscribe to Cheenta at Youtube

This site uses Akismet to reduce spam. Learn how your comment data is processed.

### Knowledge Partner  