The Annals of Statistics

Nonexistence of Informative Unbiased Estimators in Singular Problems

Richard C. Liu and Lawrence D. Brown

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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.

Article information

Ann. Statist. Volume 21, Number 1 (1993), 1-13.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F11
Secondary: 62F12: Asymptotic properties of estimators 62G05: Estimation 62A99: None of the above, but in this section

Unbiasedness modulus of continuity Hellinger distance singular problems


Liu, Richard C.; Brown, Lawrence D. Nonexistence of Informative Unbiased Estimators in Singular Problems. Ann. Statist. 21 (1993), no. 1, 1--13. doi:10.1214/aos/1176349012.

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