Nonparametric weighted estimators for biased data

Starting from a real data example in fluorescence, the problem of nonparametric estimation of a density in a biased data model is considered. Bias correction can be done in two ways: either an estimator is computed with the data and in a second time a correction (plug in estimator) is applied, or weights are directly associated with the data so that a direct estimator of the quantity of interest (weighted estimator) is obtained. In both cases, kernel and projection estimation strategies with bandwidth or model selection devices are developed. The bandwidth selection is inspired from a procedure recently proposed by Goldenshluger and Lepski (2011). Risk bounds are proved showing that the final data-driven estimators perform an automatic finite sample bias-variance tradeoff. A simulation study compares the two bias-correction methods and the different model or bandwidth selection methods. Finally real fluorescence data are studied.