Nonparametric density estimation from biased data with unknown biasing function

We present a kernel estimator for the density of a variable when sampling probabilities depend on that variable. Both the density and sampling bias weight functions are unknown and are estimated nonparametrically. To achieve this, the method requires that two independent samples be taken from a fixed finite population. An estimator of population size follows simply from our density estimator. Asymptotic bias and standard errors for these estimators are provided, and the methodology is illustrated both on simulation data and on a dual-list dataset of aboriginal people in the Vancouver-Richmond area of Canada