Nonexistence of Informative Unbiased Estimators in Singular Problems
Richard C. Liu, Lawrence D. Brown
The Annals of Statistics · 1993 · Volume 21 · Issue 1
Abstract
In many nonparametric problems, such as density estimation, nonparametric regression and so on, all the existing informative estimators are biased (asymptotic or finite sample). There has long been a suspicion that either informative unbiased estimators do not exist for such problems or they must be quite complicated. In this paper, we clarify the nonexistence of informative unbiased estimators in all singular problems both for fixed sample size and asymptotically (this includes most problems with optimal rate of convergence slower than $n^{-1/2}$). We also discuss situations in regular problems where such nonexistences can occur.
Keywords
62A99; 62F11; 62F12; 62G05; Hellinger distance; modulus of continuity; singular problems; unbiasedness;
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