absurdly unbiased estimators

“…there are important classes of problems for which the mathematics forces the existence of such estimators.”

Recently I came through a short paper written by Erich Lehmann for The American Statistician, Estimation with Inadequate Information. He analyses the apparent absurdity of using unbiased estimators or even best unbiased estimators in settings like the Poisson P(λ) observation X producing the (unique) unbiased estimator of exp(-bλ) equal to

(1-b)^x

which is indeed absurd when b>1. My first reaction to this example is that the question of what is “best” for a single observation is not very meaningful and that adding n independent Poisson observations replaces b with b/n, which gets eventually less than one. But Lehmann argues that the paradox stems from a case of missing information, as for instance in the Poisson example where the above quantity is the probability P(T=0) that T=0, when T=X+Y, Y being another unobserved Poisson with parameter (b-1)λ. In a lot of such cases, there is no unbiased estimator at all. When there is any, it must take values outside the (0,1) range, thanks to a lemma shown by Lehmann that the conditional expectation of this estimator given T is either zero or one.

I find the short paper quite interesting in exposing some reasons why the estimators cannot find enough information within the data (often a single point) to achieve an efficient estimation of the targeted function of the parameter, even though the setting may appear rather artificial.

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