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Thread: need a help on a statistics problem

  1. #1
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    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.



    please help on this one. thank you.
    Last edited by Kat-M; May 1st 2009 at 09:32 AM. Reason: hard to read without typing in tex
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  2. #2
    MHF Contributor matheagle's Avatar
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    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.
    Last edited by matheagle; May 2nd 2009 at 09:48 PM.
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