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X be random variable with distribution negative binomial NB(p,r) . r is known, 0<p<1 unknown. Find...

X be random variable with distribution negative binomial NB(p,r) . r is known, 0<p<1 unknown. Find the UMVUE of log (p)

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Answer #1

X be random variable with distribution negative binomial NB(p,r) . r is known, 0<p<1 unknown. The pmf is

We know that X is sufficient and complete for p. Let us assume the UMVUE of log(p) is g(X) where g fucntion statisfing E(g(X))= log(p). Then

Hence g(r) =0 and

This is the UMVUE of log(p).

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