1. A Study of Seven Asymmetric Kernels for the Estimation of Cumulative Distribution Functions
- Author
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Pierre Lafaye de Micheaux and Frédéric Ouimet
- Subjects
Mean squared error ,General Mathematics ,Weibull kernel ,Asymptotic distribution ,Mathematics - Statistics Theory ,inverse Gamma kernel ,Statistics Theory (math.ST) ,01 natural sciences ,Inverse Gaussian distribution ,010104 statistics & probability ,symbols.namesake ,asymptotic statistics ,0502 economics and business ,Birnbaum–Saunders kernel ,FOS: Mathematics ,Computer Science (miscellaneous) ,QA1-939 ,Applied mathematics ,Statistics::Methodology ,asymmetric kernels ,0101 mathematics ,Engineering (miscellaneous) ,050205 econometrics ,Mathematics ,Weibull distribution ,Gamma kernel ,reciprocal inverse Gaussian kernel ,Cumulative distribution function ,Probability (math.PR) ,05 social sciences ,Estimator ,62G05, 60F05, 62G20 ,nonparametric statistics ,Kernel method ,Kernel (statistics) ,symbols ,LogNormal kernel ,Mathematics - Probability ,inverse Gaussian kernel - 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., 38 pages, 2 tables, 9 figures
- Published
- 2021
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