Kernel based estimation of the distribution function for length biased data

Abstract

Empirical and kernel estimators are considered for the distribution of positive length biased data. Their asymptotic bias, variance and limiting distribution are obtained. For the kernel estimator, the asymptotically optimal bandwidth is calculated and rule of thumb bandwidths are proposed. At any point below the median, the asymptotic mean squared error of the kernel estimator is smaller than that of the empirical estimator. A suitably truncated kernel estimator is positive and we prove the strong uniform, and L2 consistency of this estimator. Simulations reveal the improved performance of the truncated kernel estimator in estimating tail probabilities based on length biased data.