How does one show that there is no unbiased estimator of λ−1\lambda^{-1} for a Poisson distribution with mean λ\lambda?

Suppose that X0,X1,,Xn are i.i.d. random variables that follow the Poisson distribution with mean λ. How can I prove that there is no unbiased estimator of the quantity 1λ?

Answer

Assume that g(X0,,Xn) is an unbiased estimator of 1/λ, that is,
(x0,,xn)Nn+10g(x0,,xn)λni=0xini=0xi!e(n+1)λ=1λ,λ>0.
Then multiplying by λe(n+1)λ and invoking the MacLaurin series of e(n+1)λ we can write the equality as
(x0,,xn)Nn+10g(x0,,xn)ni=0xi!λ1+ni=0xi=1+(n+1)λ+(n+1)2λ22+,λ>0,
where we have an equality of two power series of which one has a constant term (the right-hand side) and the other doesn’t: a contradiction. Thus no unbiased estimator exists.

Attribution
Source : Link , Question Author : billlee1231 , Answer Author : J. Virta

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