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Adaptive estimation of a density function using beta kernels

Published online by Cambridge University Press:  08 October 2014

Karine Bertin  and
Nicolas Klutchnikoff
Karine Bertin
Affiliation:
CIMFAV, Universidad de Valparaíso, Av. Pedro Montt, 2421 Valparaíso, Chile. karine.bertin@uv.cl
Nicolas Klutchnikoff
Affiliation:
CREST (ENSAI) et IRMA (UMR 7501 Université de Strasbourg et CNRS), Campus de Ker-Lann, Rue Blaise Pascal, BP 37203, 35172 BRUZ cedex, France; nicolas.klutchnikoff@ensai.fr

Abstract

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In this paper we are interested in the estimation of a density − defined on a compact interval ofℝ− from n independent andidentically distributed observations. In order to avoid boundary effect, beta kernelestimators are used and we propose a procedure (inspired by Lepski’s method) in order toselect the bandwidth. Our procedure is proved to be adaptive in an asymptotically minimaxframework. Our estimator is compared with both the cross-validation algorithm and theoracle estimator using simulated data.

Keywords

Beta kernelsadaptive estimationminimax ratesHölder spaces

Information

Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 400 - 417
Copyright
© EDP Sciences, SMAI 2014

References

Abdous, B. and Kokonendji, C.C., Consistency and asymptotic normality for discrete associated-kernel estimator. Afr. Diaspora J. Math. 8 (2009) 63–70. Google Scholar
Bertin, K. and Klutchnikoff, N., Minimax properties of beta kernel estimators. J. Statist. Plan. Inference 141 (2011) 2287–2297. Google Scholar
T. Bouezmarni and S. Van Bellegem, Nonparametric beta kernel estimator for long memory time series. Technical report (2009).
Bouezmarni, T. and Rombouts, J.V.K., Nonparametric density estimation for multivariate bounded data. J. Statist. Plann. Inference 140 (2010) 139–152. Google Scholar
Chen, S.X., Beta kernel estimators for density functions. Comput. Statist. Data Anal. 31 (1999) 131–145. Google Scholar
Chen, S.X., Beta kernel smoothers for regression curves. Statist. Sinica 10 (2000) 73–91. Google Scholar
Cline, D.B.H. and Hart, J.D., Kernel estimation of densities with discontinuities or discontinuous derivatives. Statistics 22 (1991) 69–84. Google Scholar
I. Dattner and B. Reiser, Estimation of distribution functions in measurement error models. Technical report (2010).
L. Devroye and G. Lugosi, Combinatorial methods in density estimation. Springer Series in Statistics. Springer-Verlag, New York (2001).
Giné, E. and Latała, R. and Zinn, J., Exponential and moment inequalities for U-statistics. High dimensional probability, vol. II (Seattle, WA, 1999), Birkhäuser Boston, Boston, MA. Progr. Probab. 47 (2000) 13–38. Google Scholar
Gustafsson, J., Hagmann, M., Nielsen, J.P. and Scaillet, O., Local transformation kernel density estimation of loss distributions. J. Bus. Econ. Statist. 27 (2009) 161–175. Google Scholar
Hall, P., Large sample optimality of least squares cross-validation in density estimation. Ann. Statist. 11 (1983) 1156–1174. Google Scholar
I.A. Ibragimov and R.Z. Khas’minskiĭ, More on estimation of the density of a distribution. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 108 194, 198 (1981) 72–88.
Jones, M.C., Simple boundary correction for kernel density estimation. Statist. Comput. 3 (1993) 135–146. Google Scholar
Kokonendji, C.C. and Kiessé, T.S., Discrete associated kernels method and extensions. Statist. Methodol. 8 (2011) 497–516. Google Scholar
Lejeune, M. and Sarda, P., Smooth estimators of distribution and density functions. Comput. Statist. Data Anal. 14 (1992) 457–471. Google Scholar
Lepski, O.V., Asymptotically minimax adaptive estimation. I. Upper bounds. Optimally adaptive estimates. Teor. Veroyatnost. i Primenen. 36 (1991) 645–659. Google Scholar
C. McDiarmid, On the method of bounded differences, in Surveys in combinatorics (Norwich 1989), vol. 141 of London Math. Soc. Lecture Note Ser. Cambridge University Press, Cambridge (1989) 148–188.
Müller, H.-G., Smooth optimum kernel estimators near endpoints. Biometrika 78 (1991) 521–530. Google Scholar
Renault, O. and Scaillet, O., On the way to recovery: A nonparametric bias free estimation of recovery rate densities. J. Banking and Finance 28 (2004) 2915–2931. Google Scholar
Schuster, E.F., Incorporating support constraints into nonparametric estimators of densities. Commun. Statist. − Theory Methods 14 (1985) 1123–1136. Google Scholar
B.W. Silverman, Density estimation for statistics and data analysis. Monogr. Statist. Appl. Probability. Chapman & Hall, London (1986).
Stone, Ch.J., An asymptotically optimal window selection rule for kernel density estimates. Ann. Statist. 12 (1984) 1285–1297. Google Scholar
Zhang, Sh. and Karunamuni, R.J., On kernel density estimation near endpoints. J. Statist. Plann. Inference 70 (1998) 301–316. Google Scholar
Victor de la Peña, H. and Montgomery-Smith, S.J., Decoupling inequalities for the tail probabilities of multivariate U-statistics. Ann. Probab. 23 (1995) 806–816. Google Scholar