Asymmetric kernel density estimation for biased data

Abstract

Nonparametric density estimation for nonnegative data is considered in a situation where a random sample is not directly available but the data are instead observed from the length-biased sampling. Due to the so-called boundary bias problem of the location-scale kernel, the approach in this paper is an application of asymmetric kernel. Some nonparametric density estimators are proposed. The mean integrated squared error, strong consistency, and asymptotic normality of the estimators are investigated. Simulation studies and a real data analysis illustrate the estimators.