Document Type : Original Paper
Authors
Department of Statistics, Shahid Chamran University of Ahvaz, Ahvaz,Islamic Republic of Iran
Abstract
Kernel estimation of the cumulative distribution function (CDF), when the support of the data is bounded, suffers from bias at the boundaries. To solve this problem, we introduce a new estimator for the CDF with support (0,1) based on the beta kernel function. By studying the asymptotic properties of the proposed estimator, we show that it is consistent and free from boundary bias. We conducted an extensive simulation to illustrate the performance of the proposed estimator. The results demonstrate the superiority of the proposed estimator over other commonly used estimators. As an application, we use the estimated CDF for nonparametric simulation. Using a numerical study, we show that the performance of the kernel probability density function (PDF) estimation in which a large sample simulated from the estimated CDF is employed can be noticeably improved. We also use the proposed estimator to estimate the CDF of the household health cost in Iran in 2019.
Keywords
Main Subjects
- Glivenko V. Sulla determinazione empirica delle leggi di probabilita. Gion. Ist. Ital. Attauri..1933;4:92-9.
- Cantelli FP. Sulla determinazione empirica delle leggi di probabilita. Giorn. Ist. Ital. Attuari. 1933;4(421-424).
- Azzalini A. A note on the estimation of a distribution function and quantiles by a kernel method. Biometrika. 1981 Apr 1;68(1):326-8.
- Lejeune M, Sarda P. Smooth estimators of distribution and density functions. Computational Statistics & Data Analysis. 1992 Nov 1;14(4):457-71.
- Watson GS, Leadbetter MR. Hazard analysis II. Sankhyā: The Indian Journal of Statistics, Series A. 1964 Jul 1:101-16.
- Nadaraya EA. Some new estimates for distribution functions. Theory of Probability & Its Applications. 1964;9(3):497-500.
- Mombeni HA, Mansouri B, Akhoond M. Asymmetric kernels for boundary modification in distribution function estimation. REVSTAT-Statistical Journal. 2021 Dec 2;19(4):463-84.
- Wen K, Wu X. An improved transformation-based kernel estimator of densities on the unit interval. Journal of the American Statistical Association. 2015 Apr 3;110(510):773-83.
- Tenreiro C. Boundary kernels for distribution function estimation. REVSTAT-Statistical Journal. 2013 Jun 24;11(2):169-90.
- Tenreiro C. A new class of boundary kernels for distribution function estimation. Communications in Statistics-Theory and Methods. 2018 Nov 2;47(21):5319-32.
- Koláček J, Karunamuni RJ. On boundary correction in kernel estimation of ROC curves. Austrian Journal of Statistics. 2009;38(1):17-32.
- Kolácek J, Karunamuni RJ. A generalized reflection method for kernel distribution and hazard functions estimation. Journal of Applied Probability and Statistics. 2011;6(2):73-85.
- Lafaye de Micheaux P, Ouimet F. A study of seven asymmetric kernels for the estimation of cumulative distribution functions. Mathematics. 2021 Oct 16;9(20):2605.
- Mansouri B, AtiyahSayyid Al-Farttosi S, Mombeni H, Chinipardaz R. Estimating cumulative distribution function using gamma kernel. Journal of Sciences, Islamic Republic of Iran. 2022 Mar 1;33(1):45-54.
- Mansouri, B., AtiyahSayyid Al-Farttosi S., Mombeni, H., & Chinipardaz, R. Statistical analysis and estimation of the cumulative distribution function of COVID-19 cure duration in Iraq. Journal of Statistics and Management Systems. 2022; 25(8), 2101-2112.
- Mombeni HA, Mansouri B, Akhoond M. Estimating receiver operating characteristic curve (ROC) using Birnbaum-Saunders kernel, Journal of Advanced Mathematical Modeling. 2022; (12)3:344-356.
- Chen SX. Beta kernel estimators for density functions. Computational Statistics & Data Analysis. 1999 Aug 28;31(2):131-45.
|
|
- Chen SX. Beta kernel smoothers for regression curves. StatisticaSinica. 2000 Jan 1:73-91.
- Charpentier A, Fermanian JD, Scaillet O. The estimation of copulas: Theory and practice. Copulas: From theory to application in finance. 2007:35-64.
- Bertin K, Klutchnikoff N. Minimax properties of beta kernel estimators. Journal of statistical planning and inference. 2011 Jul 1;141(7):2287-97.
- Bertin K, Klutchnikoff N. Adaptive estimation of a density function using beta kernels. ESAIM: Probability and Statistics. 2014;18:400-17.
- Igarashi G. Bias reductions for beta kernel estimation. Journal of Nonparametric Statistics. 2016 Jan 2;28(1):1-30.
- Zhang S, Karunamuni RJ. Boundary performance of the beta kernel estimators. Journal of Nonparametric Statistics. 2010 Jan 1;22(1):81-104.
- Omelka M, Gijbels I, Veraverbeke N. Improved kernel estimation of copulas: weak convergence and goodness-of-fit testing.
- Lloyd CJ. Using smoothed receiver operating characteristic curves to summarize and compare diagnostic systems. Journal of the American Statistical Association. 1998 Dec 1;93(444):1356-64.
- Pulit M. A new method of kernel-smoothing estimation of the ROC curve. Metrika. 2016 Jul;79(5):603-34.
- Duong T. Non-parametric smoothed estimation of multivariate cumulative distribution and survival functions, and receiver operating characteristic curves. Journal of the Korean Statistical Society. 2016 Mar 1;45(1):33-50.
- Simonoff JS. Smoothing methods in statistics. Springer Science & Business Media; 2012 Dec 6.
- Duong T. ks: kernel smoothing R package version 1.12. 0. http://CRAN. R-project. org/package= ks. 2021.
- Bowman A, Hall P, Prvan T. Bandwidth selection for the smoothing of distribution functions. Biometrika. 1998 Dec 1;85(4):799-808.
- Altman N, Leger C. Bandwidth selection for kernel distribution function estimation. Journal of Statistical Planning and Inference. 1995 Aug 1;46(2):195-214.
- Silverman BW. Monographs on statistics and applied probability. Density estimation for statistics and data analysis. 1986;26.
- Bouezmarni T, Scaillet O. Consistency of asymmetric kernel density estimators and smoothed histograms with application to income data. Econometric Theory. 2005 Apr;21(2):390-412.
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