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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.


Similar Problems and Solutions



ISI MStat PSB 2008 Problem 10
Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

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 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.


Similar Problems and Solutions



ISI MStat PSB 2008 Problem 10
Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

Subscribe to Cheenta at Youtube


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