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Published October 16, 2021 | Published + Submitted
Journal Article Open

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

Abstract

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.

Additional Information

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). Received: 15 September 2021; Accepted: 12 October 2021; Published: 16 October 2021. We thank Benedikt Funke for reminding us of the representation min{T₁,T₂} = 1/2(T₁+T₂) − 1/2|T₁−T₂|, which helped tightening up the MSE and MISE results in Section 6 and Section 7. This research includes computations using the computational cluster Katana supported by Research Technology Services at UNSW Sydney. We thank the referees for their insightful remarks which led to improvements in the presentation of this paper. F. Ouimet is supported by postdoctoral fellowships from the NSERC (PDF) and the FRQNT (B3X supplement and B3XR). Author Contributions: Conceptualisation, methodology, investigation, formal analysis, software, coding and simulations, validation, visualisation, writing—original draft preparation, writing—review and editing, review of the literature, theoretical results and proofs, F.O.; software, coding and simulations, validation, P.L.d.M. All authors have read and agreed to the published version of the manuscript. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The R code for the simulations in Section 8 is available online.

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Published - mathematics-09-02605.pdf

Submitted - 2011.14893.pdf

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Created:
August 22, 2023
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