A Study of Seven Asymmetric Kernels for the Estimation of Cumulative Distribution Functions

Pierre Lafaye de Micheaux (1, 2, 3) , Frédéric Ouimet (4, 5)

1 UPVM - Université Paul-Valéry - Montpellier 3
2 IDESP - Institut Desbrest de santé publique
3 UNSW - University of New South Wales [Sydney]
4 McGill University = Université McGill [Montréal, Canada]
5 CALTECH - California Institute of Technology

In this paper, we complement a study recently conducted in a paper of H.A. Mombeni, B. Masouri and M.R. Akhoond by introducing five new asymmetric kernel c.d.f. estimators on the half-line [0,∞), 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, Masouri and Akhoond. 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 from C. Tenreiro.

Mots clés: Asymmetric kernels; Asymptotic statistics; Nonparametric statistics; Gamma Kernel; Inverse Gamma kernel; LogNormal kernel; Inverse Gaussian kernel; Reciprocal inverse Gaussian kernel; Birnbaum–Saunders kernel; Weibull kernel

Domaines: Statistiques [math.ST]

Pierre Lafaye de Micheaux, Frédéric Ouimet. A Study of Seven Asymmetric Kernels for the Estimation of Cumulative Distribution Functions. Mathematics , 2021, 9 (20), pp.2605. ⟨10.3390/math9202605⟩. ⟨hal-03675741⟩