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Math Help - need a help on a statistics problem

  1. #1
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    need a help on a statistics problem

    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.
    Last edited by Kat-M; May 1st 2009 at 10:32 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 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.
    Last edited by matheagle; May 2nd 2009 at 10:48 PM.
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