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In Mombeni et al. (2019), Birnbaum-Saunders and Weibull kernel estimatorswere introduced for the estimation of cumulative distribution functions(c.d.f.s) supported on the half-line $[0,\infty)$. They were the first authorsto use asymmetric kernels in the context of c.d.f. estimation. Their estimatorswere shown to perform better numerically than traditional methods such as thebasic kernel method and the boundary modified version from Tenreiro (2013). Inthe present paper, we complement their study by introducing five new asymmetrickernel c.d.f. estimators, namely the Gamma, inverse Gamma, lognormal, inverseGaussian and reciprocal inverse Gaussian kernel c.d.f. estimators. For thesefive new estimators, we prove the asymptotic normality and we find asymptoticexpressions for the following quantities: bias, variance, mean squared errorand mean integrated squared error. A numerical study then compares theperformance of the five new c.d.f. estimators against traditional methods andthe 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 andBirnbaum-Saunders kernel c.d.f. estimators perform the best overall, while theother asymmetric kernel estimators are sometimes better but always at leastcompetitive against the boundary kernel method.

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