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    Poisson Sufficiency and Unbiased Estimators

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    From a Poisson (θ) distribution a random sample X1, X2, ... , Xn is selected. Given that that S = X1 + ... + Xn is sufficient for θ and also has the Poisson(nθ) distribution, we can define gr,k(s) by
    gr,k(s) = {s!/(s - r)!} n-r {1 - (k/n)}s-r, s = r, r + 1, ... , 0 otherwise,

    in which r = 0, 1, 2, ...
    and k is a real constant.

    (i) Show that E[gr,k(S)] = θ^r exp(- kθ) = τr,k(θ).

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    Solution Summary

    The solution contains the determination of a sufficient statistic for a Poisson parameter and obtaining an unbiased estimator for a function of the parameter.