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June 2004 Density estimation for biased data
Sam Efromovich
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Ann. Statist. 32(3): 1137-1161 (June 2004). DOI: 10.1214/009053604000000300


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.


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Sam Efromovich. "Density estimation for biased data." Ann. Statist. 32 (3) 1137 - 1161, June 2004.


Published: June 2004
First available in Project Euclid: 24 May 2004

zbMATH: 1091.62022
MathSciNet: MR2065200
Digital Object Identifier: 10.1214/009053604000000300

Primary: 625G07
Secondary: 62C05 , 62E20

Keywords: Adaptation , average risk , coefficient of difficulty , nonparametric , restricted minimax , small sample

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 3 • June 2004
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