Beta Kernel Estimator for a Cumulative Distribution Function with Bounded Support

Abstract

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.

Keywords

• nonparametric estimation
• kernel estimator
• boundary bias
• bootstrap
• household cost