Transformation-based density estimation for weighted distributions

In this paper we consider the estimation of a density f on the basis of ran dom sample from a weighted distribution G with density g given by g(x) = w(x)f(x)/mu w, where w(u) > 0 for all u and mu(w) = integral w(u)f(u)du < infinity. 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 e stimating the density f. The approach is motivated by the form of the nonpa rametric estimator maximum likelihood off in the same context and under a m onotonicity constraint. Since the method does not depend on the specific de nsity estimate used (only the transformation), it can be used to construct both simple density estimates (histograms or frequency polygons) and more c omplex methods with favorable properties (e.g., local or penalized likeliho od estimates). Monte Carlo simulations indicate that transformation-based d ensity 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.