The Annals of Statistics

Density estimation for biased data

Sam Efromovich

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The concept of biased data is well known and its practical applications range from social sciences and biology to economics and quality control. These observations arise when a sampling procedure chooses an observation with probability that depends on the value of the observation. This is an interesting sampling procedure because it favors some observations and neglects others. It is known that biasing does not change rates of nonparametric density estimation, but no results are available about sharp constants. This article presents asymptotic results on sharp minimax density estimation. In particular, a coefficient of difficulty is introduced that shows the relationship between sample sizes of direct and biased samples that imply the same accuracy of estimation. The notion of the restricted local minimax, where a low-frequency part of the estimated density is known, is introduced; it sheds new light on the phenomenon of nonparametric superefficiency. Results of a numerical study are presented.

Article information

Ann. Statist. Volume 32, Number 3 (2004), 1137-1161.

First available in Project Euclid: 24 May 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 625G07
Secondary: 62C05: General considerations 62E20: Asymptotic distribution theory

Adaptation average risk coefficient of difficulty nonparametric restricted minimax small sample


Efromovich, Sam. Density estimation for biased data. Ann. Statist. 32 (2004), no. 3, 1137--1161. doi:10.1214/009053604000000300.

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