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.
Mots clés : frontier estimation, kernel estimation, strong uniform consistency, Hall class
@article{PS_2014__18__642_0,
author = {Girard, St\'ephane and Guillou, Armelle and Stupfler, Gilles},
title = {Uniform strong consistency of a frontier estimator using kernel regression on high order moments},
journal = {ESAIM: Probability and Statistics},
pages = {642--666},
publisher = {EDP-Sciences},
volume = {18},
year = {2014},
doi = {10.1051/ps/2013050},
mrnumber = {3049899},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2013050/}
}
TY - JOUR AU - Girard, Stéphane AU - Guillou, Armelle AU - Stupfler, Gilles TI - Uniform strong consistency of a frontier estimator using kernel regression on high order moments JO - ESAIM: Probability and Statistics PY - 2014 SP - 642 EP - 666 VL - 18 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2013050/ DO - 10.1051/ps/2013050 LA - en ID - PS_2014__18__642_0 ER -
%0 Journal Article %A Girard, Stéphane %A Guillou, Armelle %A Stupfler, Gilles %T Uniform strong consistency of a frontier estimator using kernel regression on high order moments %J ESAIM: Probability and Statistics %D 2014 %P 642-666 %V 18 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2013050/ %R 10.1051/ps/2013050 %G en %F PS_2014__18__642_0
Girard, Stéphane; Guillou, Armelle; Stupfler, Gilles. Uniform strong consistency of a frontier estimator using kernel regression on high order moments. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 642-666. doi : 10.1051/ps/2013050. http://www.numdam.org/articles/10.1051/ps/2013050/
[1] and , Handbook of Mathematical Functions. Dover (1965).
[2] , and , Nonparametric frontier estimation: a conditional quantile-based approach. Econom. Theory 21 (2005) 358-389. | MR | Zbl
[3] , and . Regular Variation. Cambridge, Cambridge University Press (1987). | MR | Zbl
[4] , and , Nonparametric frontier estimation: a robust approach. J. Econom. 106 (2002) 1-25. | MR | Zbl
[5] and , Robust nonparametric estimators of monotone boundaries. J. Multivariate Anal. 96 (2005) 311-331. | MR | Zbl
[6] , and , 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.
[7] and , An empirical process approach to the uniform consistency of kernel-type function estimators. J. Theoret. Probab. 13 (2000) 1-37. | MR | Zbl
[8] , and , Modelling extremal events. Springer (1997). | MR
[9] and , Estimation of the conditional tail-index using a smoothed local Hill estimator. Extremes 17 (2014) 45-75. | MR
[10] , Sur un problème d'estimation géométrique. Publ. Inst. Statist. Univ. Paris XIII (1964) 191-210. | MR | Zbl
[11] , and , Asymptotic normality of the L1 −error of a boundary estimator. Nonparametr. Stat. 18 (2006) 21-31. | MR | Zbl
[12] , and , Frontier estimation with kernel regression on high order moments. J. Multivariate Anal. 116 (2013) 172-189. | MR | Zbl
[13] , and , L1 −optimal nonparametric frontier estimation via linear programming. Autom. Remote Control 66 (2005) 2000-2018. | MR | Zbl
[14] and , Frontier estimation via kernel regression on high power-transformed data. J. Multivariate Anal. 99 (2008) 403-420. | MR | Zbl
[15] and , Frontier estimation with local polynomials and high power-transformed data. J. Multivariate Anal. 100 (2009) 1691-1705. | MR | Zbl
[16] , On estimating the endpoint of a distribution. Ann. Statist. 10 (1982) 556-568. | MR | Zbl
[17] , and , Strong uniform consistency rates for estimators of conditional functionals. Ann. Statist. 16 (1988) 1428-1449. | MR | Zbl
[18] and , Optimal bandwidth selection in nonparametric regression function estimation. Ann. Statist. 13 (1985) 1465-1481. | MR | Zbl
[19] , and , Estimation of non-sharp support boundaries. J. Multivariate Anal. 55 (1995) 205-218. | MR | Zbl
[20] , Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 (1963) 13-30. | MR | Zbl
[21] and , Estimating the edge of a Poisson process by orthogonal series. J. Statist. Plann. Inference 46 (1995) 215-234. | MR | Zbl
[22] , and , Efficient estimation of monotone boundaries. Ann. Statist. 23 (1995) 476-489. | MR | Zbl
[23] , and , Asymptotic properties of a conditional quantile estimator with randomly truncated data. J. Multivariate Anal. 100 (2009) 546-559. | MR | Zbl
[24] and , Weak and strong uniform consistency of kernel regression estimates. Z. Wahrscheinlichkeitstheorie verw. Gebiete 61 (1982) 405-415. | MR | Zbl
[25] , On non-parametric estimates of density functions and regression curves. Theory Probab. Appl. 10 (1965) 186-190. | MR | Zbl
[26] , On estimation of a probability density function and mode. Ann. Math. Statist. 33 (1962) 1065-1076. | MR | Zbl
[27] , Remarks on some nonparametric estimates of a density function. Ann. Math. Statist. 27 (1956) 832-837. | MR | Zbl
[28] , Weak and strong uniform consistency of the kernel estimate of a density and its derivatives. Ann. Statist. 6 (1978) 177-184. | MR | Zbl
[29] , A law of the iterated logarithm for kernel density estimators. Ann. Probab. 10 (1982) 414-422. | MR | Zbl
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