need a help on a statistics problem

**Kat-M** · May 1st 2009, 01:12 AM

Let $X1,X2,...Xn$ be a random sample from Geometric Distrubution with parameter $p$ , that is $f(x;p)=p(1-p)^{(x-1)}$

Find the Minimum Variance Unbiased Estimator for $p[X=c]$ , where $c$ is a known positive interger.

please help on this one. thank you.

**matheagle** · May 1st 2009, 05:28 PM

You first find the suff stat, which is any multiple of $S_n=\sum_{i=1}^n X_i$ .

Then you want to find an unbiased estimator of your 'parameter' that is based on this sum.

The thing you want to estimate is $p(1-p)^{c-1}$ .

The expected value of S is n/p. You can try S^c and see what you get, but the first thing I would note is that
S is a sum of geo's, hence it's a negative binomial.

To use the Rao-Blackwell Theorem directly let $T= I(X_1=c)$ , so T is unbiased for that probability.

The UMVUE will be $E(T|S_n=s)$ .

See example 9.1 from http://www.stat.unc.edu/faculty/cji/lecture9.pdf for the messy next steps.

So you need $E(I(X_1=c)|S_n=s)=P(X_1=c|X_1+\cdots +X_n=s)$

$= {P(X_1=c, X_2+\cdots +X_n=s-c)\over P(X_1+\cdots +X_n=s)} ={P(X_1=c)P(X_2+\cdots +X_n=s-c)\over P(X_1+\cdots +X_n=s)}$

We know $P(X_1=c)$ since X_1 is a geo.

And $P(X_2+\cdots +X_n=s-c)$ via $S_{n-1}$ is a NB where you are waiting for the $(n-1)^{th}$ success.

$P(X_1+\cdots +X_n=s)$ via $S_n$ is a NB where you are waiting for the $n^{th}$ success.

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