Statistics | Part IB, 2004

State and prove the Rao-Blackwell Theorem.

Suppose that X1,X2,,XnX_{1}, X_{2}, \ldots, X_{n} are independent, identically-distributed random variables with distribution

P(X1=r)=pr1(1p),r=1,2,P\left(X_{1}=r\right)=p^{r-1}(1-p), \quad r=1,2, \ldots

where p,0<p<1p, 0<p<1, is an unknown parameter. Determine a one-dimensional sufficient statistic, TT, for pp.

By first finding a simple unbiased estimate for pp, or otherwise, determine an unbiased estimate for pp which is a function of TT.

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