In this paper, we prove the strong uniform consistency and asymptotic normality of the Kernel Density Estimator proposed by Jones for length-biased data. The approach is based on the invariance principle for the empirical processes proved by Horvath. All simulations are drawn for different cases to demonstrate both, consistency and asymptotic normality and the method is illustrated by real automobile brake pads data.

Kernel Density Estimators are the basic tools for Density estimation in non-parametric statistics. The k-nearest neighbor Kernel Estimators represent a special form of Kernel Density Estimators, in which the bandwidth is varied depending on the location of the sample points. In this paper, we initially introduce the k-nearest neighbor Kernel Density Estimator in the random left-truncation model, and then prove some of its asymptotic behaviors, such as strong uniform consistency and asymptotic normality. In particular, we show that the proposed Estimator has truncation-free variance. Simulations are presented to illustrate the results and show how the Estimator behaves for finite samples. Moreover, the proposed Estimator is used to estimate the Density function of a real data set.

Length-biased data are widely seen in applications. They are mostly applicable in epidemiological studies or survival analysis in medical researches. Here we aim to propose a Berry-Esseen type bound for the Kernel Density Estimator of this kind of data. The rate of normal convergence in the proposed Berry-Esseen type theorem is shown to be O(n^(-1/6) ) modulo logarithmic term as n tends to infinity by a proper choice of the bandwidth. The results of a simulation study is also presented in this paper inorder to examine the performance of the result.

The receiver operating characteristic curve is a simple graphical tool used to assess the accuracy of diagnostics tests‎. ‎Pulit (2016) proposed an innovative approach for estimating the receiver operating characteristic curves based on Kernel smoothing‎. ‎Although his proposed Estimator is highly appealing in several aspects‎, ‎it suffers from the well-known boundary bias effect‎. ‎In this paper‎, ‎we highlight this drawback and propose a new modified Estimator that uses an appropriate boundary Kernel‎. ‎The asymptotic convergence of the proposed Estimator at boundary points is demonstrated‎. ‎Using both simulated and real data sets‎, ‎we illustrate the performance of the proposed Estimator‎. ‎The results show that the proposed Estimator outperforms not only the Pulit’s Estimator but also other commonly used Estimators‎.

Kernel estimation of the cumulative distribution function (CDF), when the support of the data is bounded, suffers from bias at the boundaries. To solve this problem, we introduce a new Estimator for the CDF with support (0,1) based on the beta Kernel function. By studying the asymptotic properties of the proposed Estimator, we show that it is consistent and free from boundary bias. We conducted an extensive simulation to illustrate the performance of the proposed Estimator. The results demonstrate the superiority of the proposed Estimator over other commonly used Estimators. As an application, we use the estimated CDF for nonparametric simulation. Using a numerical study, we show that the performance of the Kernel probability Density function (PDF) estimation in which a large sample simulated from the estimated CDF is employed can be noticeably improved. We also use the proposed Estimator to estimate the CDF of the household health cost in Iran in 2019.