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# Poisson Sufficiency and Unbiased Estimators

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From a Poisson (&#952;) distribution a random sample X1, X2, ... , Xn is selected. Given that that S = X1 + ... + Xn is sufficient for &#952; and also has the Poisson(n&#952;) 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)] = &#952;^r exp(- k&#952;) = &#964;r,k(&#952;).

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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.

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