Abstract
In a regression context where a response variable Y ∈ ℝ is recorded with a covariate X ∈ ℝ p, two situations can occur simultaneously: (a) we are interested in the tail of the conditional distribution and not on the central part of the distribution and (b) the number p of regressors is large. To our knowledge, these two situations have only been considered separately in the literature. The aim of this paper is to propose a new dimension reduction approach adapted to the tail of the distribution in order to propose an efficient conditional extreme quantile estimator when the dimension p is large. The results are illustrated on simulated data and on a real dataset.
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References
Araújo Santos, P., Fraga Alves, M.I., Gomes, M.I.: Peaks over random threshold methodology for tail index and high quantile estimation. REVSTAT 4(3), 227–247 (2006)
Basu, D., Pereira, C.A.B.: Conditional independence in statistics. Sankhyā 45(3), Series A, 324–337 (1983)
Chaudhuri, P.: Nonparametric estimates of regression quantiles and their local Bahadur representation. Ann. Stat. 19(2), 760–777 (1991)
Cook, R.D.: On the interpretation of regression plots. J. Am. Stat. Assoc. 89(425), 177–189 (1994)
Cook, R.D., Forzani, L.: Likelihood-based sufficient dimension reduction. J. Am. Stat. Assoc. 104, 197–208 (2009)
Cook, R.D., Weisberg, S.: Discussion of sliced inverse regression for dimension reduction. J. Am. Stat. Assoc. 86, 328–332 (1991)
Daouia, A., Gardes, L., Girard, S.: On Kernel smoothing for extremal quantile regression. Bernoulli 19(5B), 2557–2589 (2013)
Daouia, A., Gardes, L., Girard, S., Lekina, A.: Kernel estimators of extreme level curves. Test 20(2), 311–333 (2011)
Davison, A.C., Smith, R.L.: Models for exceedances over high thresholds. J. R. Stat. Soc. Ser. B 52, 393–442 (1990)
Davison, A.C., Ramesh, N.I.: Local likelihood smoothing of sample extremes. J. R. Stat. Soc. Ser. B 62, 191–208 (2000)
de Haan, L., Ferreira, A.: Extreme Value Theory: An Introduction. Springer (2006)
Eastoe, E.F., Tawn, J.A.: Modelling non-stationary extremes with application to surface level ozone. J. R. Stat. Soc. Ser. C 58, 25–45 (2009)
Fisher, R.A., Tippett, L.H.C.: Limiting forms of the frequency distribution of the largest or smallest member of a sample. Proc. Camb. Philos. Soc. 24, 180–190 (1928)
Fraga Alves, M.I., Gomes, M.I., de Haan, L., Neves, C.: A note on second order conditions in extreme value theory: linking general and heavy tail conditions. REVSTAT - Stat. J. 5(3), 285–304 (2007)
Gannoun, A., Girard, S., Guinot, C., Saracco, J.: Sliced inverse regression in reference curves estimation. Comput. Stat. Data Anal. 46(1), 103–122 (2004)
Gardes, L.: A general estimator for the extreme value index: applications to conditional and heteroscedastic extremes. Extremes 18(3), 479–510 (2015)
Gardes, L., Girard, S.: Conditional extremes from heavy-tailed distributions: an application to the estimation of extreme rainfall return levels. Extremes 13(2), 177–204 (2010)
Gardes, L., Girard, S.: Functional kernel estimators of large conditional quantiles. Electron. J. Stat. 6, 1715–1744 (2012)
Gardes, L., Girard, S.: On the estimation of the functional Weibull tail-coefficient. J. Multivar. Anal. 146, 29–45 (2016)
Gasser, T., Müller, H.G.: Estimating regression functions and their derivatives by the kernel method. Scand. J. Stat. 11, 171–185 (1984)
Gnedenko, B.V.: Sur la distribution limite du terme maximum d’une série aléatoire. Ann. Math. 44, 423–453 (1943)
Hooke, R., Jeeves, T.A.: Direct search solution of numerical and statistical problems. J. ACM 8(2), 212–229 (1961)
Han, S., Bian, H., Feng, Y., Liu, A., Li, Zeng, F., Zhang, X: Analysis of the relationship between O3,, NO and NO2 in Tianjin, China. Aerosol Air Qual. Res. 11, 128–139 (2011)
Ichimura, H.: Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. J. Econ. 58, 71–120 (1993)
Koenker, R., Basset, G.S.: Regression quantiles. Econometrica 46, 33–50 (1978)
Li, K.C.: Sliced inverse regression for dimension reduction. J. Am. Stat. Assoc. 86, 316–327 (1991)
Li, K.C.: On principal hessian directions for data visualization and dimension reduction: another application of Stein’s lemma. J. Am. Stat. Assoc. 87, 1025–1039 (1992)
Nadaraya, E.A.: On estimating regression. Theory Probab. Appl. 9(1), 141–142 (1964)
Northrop, P.J., Jonathan, P.: Threshold modelling of spatially dependent non-stationary extremes with application to hurricane-induced wave heights. Environmetrics 22(7), 799–809 (2011)
Parzen, E.: On the estimation of a probability density function and mode. Ann. Math. Stat. 33, 1065–1076 (1962)
Russell, B.T., Cooley, D., Porter, W.C., Reich, B.J., Heald, C.L.: Data mining for extreme behavior with application to ground level ozone. Ann. Appl. Stat. 10, 1673–1698 (2016)
Scrucca, L.: Model-based SIR for dimension reduction. Comput. Stat. Data Anal. 55, 3010–3026 (2011)
Wang, E., Tsai, C.: Tail index regression. J. Am. Stat. Assoc. 104(487), 1233–1240 (2009)
Watson, G.S.: Smooth regression analysis. Sankhyā 26(15), 175–184 (1964)
Weisberg, W.: Dimension reduction regression in R. J. Stat. Softw. 7, 1–22 (2002)
Wu, T.Z., Yu, K., Yu, Y.: Single-index quantile regression. J. Multivar. Anal. 101, 1607–1621 (2010)
Xia, Y.: Model checking in regression via dimension reduction. Biometrika 96(1), 133–148 (2009)
Yu, K., Jones, M.C.: Local linear quantile regression. J. Am. Stat. Assoc. 93(441), 228–237 (1998)
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Gardes, L. Tail dimension reduction for extreme quantile estimation. Extremes 21, 57–95 (2018). https://doi.org/10.1007/s10687-017-0300-x
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DOI: https://doi.org/10.1007/s10687-017-0300-x