Transformation-Based Density Estimation for Weighted Distributions

In this paper we consider the estimation of a density f on the basis of random sample from a weighted distribution G with density g given by g(x) = w(x)f(x)/ Atilde; Acirc;micro;w, where w(u)gt;0 for all u and Atilde; Acirc;micro;w = Atilde;cent;Acirc; Acirc;laquo; w(u)f(u)du lt; Atilde;cent;Acirc; Acirc; . A special case of this situation is that of length-biased sampling, where w(x) = x. In this paper we examine a simple transformation-based approach to estimating the density f. The approach is motivated by the form of the nonparametric estimator of f in the same context and under a monotonicity constraint. Since the method does not depend on the specific density estimate used (only the transformation), it can be used to construct both simple density estimates (histograms or frequency polygons) and more complex methods with favorable properties (e.g., local or penalized likelihood estimates). Monte Carlo simulations indicate that transformation-based density estimation can outperform the kernel-based estimator of Jones (1991) depending on the weight function w, and leads to much better estimation of monotone densities than the nonparametric maximum likelihood estimator