Distribution estimation for biased data

Biased 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 that favors some observations and neglects others. This article presents results on sharp minimax estimation of nonparametric distribution functions and densities that allow us to answer the long-standing Cox question on how a biased sampling affects accuracy of the estimation. The theory has many interesting byproducts including optimal results for a boundary problem, a model of recurrence distributions and an average risk. The case of small data sets is also discussed, numerical simulations are presented and S-PLUS software with adaptive estimates is available. (C) 2003 Elsevier B.V. All rights reserved.