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Uniform strong consistency of a frontier estimatorusing kernel regression on high order moments

Published online by Cambridge University Press:  15 October 2014

Stéphane Girard ,
Armelle Guillou  and
Gilles Stupfler
Stéphane Girard
Affiliation:
Team Mistis, INRIA Rhône-Alpes & LJK, Inovallée, 655, av. de l’Europe, Montbonnot, 38334 Saint-Ismier cedex, France. [email protected]
Armelle Guillou
Affiliation:
Université de Strasbourg & CNRS, IRMA, UMR 7501, 7 rue René Descartes, 67084 Strasbourg cedex, France
Gilles Stupfler
Affiliation:
Aix Marseille Université, CERGAM, EA 4225, 15-19 allée Claude Forbin, 13628 Aix-en-Provence cedex 1, France
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Abstract

We consider the high order moments estimator of the frontier of a random pair, introduced by [S. Girard, A. Guillou and G. Stupfler, J. Multivariate Anal. 116 (2013) 172–189]. In the present paper, we show that this estimator is strongly uniformly consistent on compact sets and its rate of convergence is given when the conditional cumulative distribution function belongs to the Hall class of distribution functions.

Keywords

Frontier estimationkernel estimationstrong uniform consistencyHall class
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 642 - 666
Copyright
© EDP Sciences, SMAI, 2014

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References

M. Abramovitz and I. Stegun, Handbook of Mathematical Functions. Dover (1965).
Aragon, Y., Daouia, A. and Thomas-Agnan, C., Nonparametric frontier estimation: a conditional quantile-based approach. Econom. Theory 21 (2005) 358–389. Google Scholar
N.H. Bingham, C.M. Goldie and J.L. Teugels. Regular Variation. Cambridge, Cambridge University Press (1987).
Cazals, C., Florens, J.-P. and Simar, L., Nonparametric frontier estimation: a robust approach. J. Econom. 106 (2002) 1–25. Google Scholar
Daouia, A. and Simar, L., Robust nonparametric estimators of monotone boundaries. J. Multivariate Anal. 96 (2005) 311–331. Google Scholar
D. Deprins, L. Simar and H. Tulkens, Measuring labor efficiency in post offices, in The Performance of Public Enterprises: Concepts and Measurements. Edited by P. Pestieau, M. Marchand and H. Tulkens. Amsterdam: North Holland (1984) 243–267.
Einmahl, U. and Mason, D.M., An empirical process approach to the uniform consistency of kernel-type function estimators. J. Theoret. Probab. 13 (2000) 1–37. Google Scholar
P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling extremal events. Springer (1997).
Gardes, L. and Stupfler, G., Estimation of the conditional tail-index using a smoothed local Hill estimator. Extremes 17 (2014) 45–75. Google Scholar
Geffroy, J., Sur un problème d’estimation géométrique. Publ. Inst. Statist. Univ. Paris XIII (1964) 191–210. Google Scholar
Geffroy, J., Girard, S. and Jacob, P., Asymptotic normality of the L 1 −error of a boundary estimator. Nonparametr. Stat. 18 (2006) 21–31. Google Scholar
Girard, S., Guillou, A. and Stupfler, G., Frontier estimation with kernel regression on high order moments. J. Multivariate Anal. 116 (2013) 172–189. Google Scholar
Girard, S., Iouditski, A. and Nazin, A., L 1 −optimal nonparametric frontier estimation via linear programming. Autom. Remote Control 66 (2005) 2000–2018. Google Scholar
Girard, S. and Jacob, P., Frontier estimation via kernel regression on high power-transformed data. J. Multivariate Anal. 99 (2008) 403–420. Google Scholar
Girard, S. and Jacob, P., Frontier estimation with local polynomials and high power-transformed data. J. Multivariate Anal. 100 (2009) 1691–1705. Google Scholar
Hall, P., On estimating the endpoint of a distribution. Ann. Statist. 10 (1982) 556–568. Google Scholar
Härdle, W., Janssen, P. and Serfling, R., Strong uniform consistency rates for estimators of conditional functionals. Ann. Statist. 16 (1988) 1428–1449. Google Scholar
Härdle, W. and Marron, J.S., Optimal bandwidth selection in nonparametric regression function estimation. Ann. Statist. 13 (1985) 1465–1481. Google Scholar
Härdle, W., Park, B.U. and Tsybakov, A., Estimation of non-sharp support boundaries. J. Multivariate Anal. 55 (1995) 205–218. Google Scholar
Hoeffding, W., Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 (1963) 13–30. Google Scholar
Jacob, P. and Suquet, C., Estimating the edge of a Poisson process by orthogonal series. J. Statist. Plann. Inference 46 (1995) 215–234. Google Scholar
Korostelev, A., Simar, L. and Tsybakov, A., Efficient estimation of monotone boundaries. Ann. Statist. 23 (1995) 476–489. Google Scholar
Lemdani, M., Ould-Saïd, E. and Poulin, N., Asymptotic properties of a conditional quantile estimator with randomly truncated data. J. Multivariate Anal. 100 (2009) 546–559. Google Scholar
Mack, Y.P. and Silverman, B.W., Weak and strong uniform consistency of kernel regression estimates. Z. Wahrscheinlichkeitstheorie verw. Gebiete 61 (1982) 405–415. Google Scholar
Nadaraya, E.A., On non-parametric estimates of density functions and regression curves. Theory Probab. Appl. 10 (1965) 186–190. Google Scholar
Parzen, E., On estimation of a probability density function and mode. Ann. Math. Statist. 33 (1962) 1065–1076. Google Scholar
Rosenblatt, M., Remarks on some nonparametric estimates of a density function. Ann. Math. Statist. 27 (1956) 832–837. Google Scholar
Silverman, B.W., Weak and strong uniform consistency of the kernel estimate of a density and its derivatives. Ann. Statist. 6 (1978) 177–184. Google Scholar
Stute, W., A law of the iterated logarithm for kernel density estimators. Ann. Probab. 10 (1982) 414–422. Google Scholar