# need a help on a statistics problem

• May 1st 2009, 01:12 AM
Kat-M
need a help on a statistics problem
Let $\displaystyle X1,X2,...Xn$ be a random sample from Geometric Distrubution with parameter $\displaystyle p$, that is $\displaystyle f(x;p)=p(1-p)^{(x-1)}$

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

• May 1st 2009, 05:28 PM
matheagle
You first find the suff stat, which is any multiple of $\displaystyle S_n=\sum_{i=1}^n X_i$.

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

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

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 $\displaystyle T= I(X_1=c)$, so T is unbiased for that probability.

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

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

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

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

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

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

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