A study of seven asymmetric kernels for the estimation of cumulative distribution functions

TL;DR

This paper introduces five new asymmetric kernel estimators for cumulative distribution functions supported on [0,∞), proves their asymptotic properties, and compares their performance with existing methods through numerical analysis.

Contribution

It extends previous work by adding five novel asymmetric kernel estimators and analyzing their asymptotic behavior and performance in c.d.f. estimation.

Findings

Lognormal and Birnbaum-Saunders estimators perform best overall.

Other asymmetric kernels are competitive with boundary methods.

New estimators have proven asymptotic normality and explicit error expressions.

Abstract

In Mombeni et al. (2019), Birnbaum-Saunders and Weibull kernel estimators were introduced for the estimation of cumulative distribution functions (c.d.f.s) supported on the half-line $[0,\infty)$. They were the first authors to use asymmetric kernels in the context of c.d.f. estimation. Their estimators were shown to perform better numerically than traditional methods such as the basic kernel method and the boundary modified version from Tenreiro (2013). In the present paper, we complement their study by introducing five new asymmetric kernel c.d.f. estimators, namely the Gamma, inverse Gamma, lognormal, inverse Gaussian and reciprocal inverse Gaussian kernel c.d.f. estimators. For these five new estimators, we prove the asymptotic normality and we find asymptotic expressions for the following quantities: bias, variance, mean squared error and mean integrated squared error. A numerical study then compares the performance of the five new c.d.f. estimators against traditional methods and the Birnbaum-Saunders and Weibull kernel c.d.f. estimators from Mombeni et al. (2019). By using the same experimental design, we show that the lognormal and Birnbaum-Saunders kernel c.d.f. estimators perform the best overall, while the other asymmetric kernel estimators are sometimes better but always at least competitive against the boundary kernel method.