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 \(L_2\) consistency of this estimator. Simulations reveal the improved performance of the truncated kernel estimator in estimating tail probabilities based on length biased data.

经验和核估计器被考虑用于正长度偏差数据的分布。获得了它们的渐近偏差、方差和极限分布。对于核估计器，计算渐近最优带宽并提出经验法则带宽。在中位数以下的任何点，核估计器的渐近均方误差都小于经验估计器的渐近均方误差。一个适当截断的核估计器是正的，我们证明了这个估计器的强一致和\(L_2\)一致性。模拟揭示了截断核估计器在基于长度偏差数据估计尾概率方面的改进性能。