Functional kernel estimation of the conditional extreme value index under random right censoring. (English. French summary) Zbl 1485.62060
Summary: Estimation of the extreme-value index of a heavy-tailed distribution is investigated when some functional random covariate (i.e. valued in some infinite dimensional space) information is available and the scalar response variable is right-censored. A weighted kernel version of Hill’s estimator of the extreme-value index is proposed and its asymptotic normality is established under mild assumptions. A simulation study is conducted to assess the finite-sample behavior of the proposed estimator. An application to ambulatory blood pressure trajectories and clinical outcome in stroke patients is also provided.
MSC:
| 62G32 | Statistics of extreme values; tail inference |
| 62G07 | Density estimation |
| 62N02 | Estimation in survival analysis and censored data |
| 62P10 | Applications of statistics to biology and medical sciences; meta analysis |
| 62R10 | Functional data analysis |
Keywords:
censored data; conditional extreme value index; heavy-tailed distributions; functional data; kernel estimatorReferences:
| [1] | BEIRLANT, J., GUILLOU, A., DIERCKX, G., AND FILS-V ILLETARD , A. (2007). Estimation of the extreme value index and extreme quantiles under random censoring. Extremes, 10 (3), 151-1174. · Zbl 1157.62027 |
| [2] | BEIRLANT, J., GUILLOU, A. AND TOULEMONDE, G. (2010). Peaks-over-threshold model-ing under random censoring. Communications in Statistics-Theory and Methods,39(7), 1158-1179. · Zbl 1284.62612 |
| [3] | BEIRLANT, J., MARIBE, G., AND VERSTER, A. (2018). Penalized bias reduction in extreme value estimation for censored Pareto-type data, and long-tailed insurance applications. Insurance: Mathematics and Economics, 78, 114-122. · Zbl 1396.62096 |
| [4] | BEIRLANT, J., WORMS, J., AND WORMS, R. (2019). Estimation of the extreme value index in a censorship framework: Asymptotic and finite sample behavior. Journal of Statistical Planning and Inference, 202, 31-56. · Zbl 1418.62214 |
| [5] | BOSQ , D. (2000). Linear processes in function spaces: theory and applications. Journal of the Royal Statistical Society. Series B (Methodological). · Zbl 0962.60004 |
| [6] | BRAHIMI, B., MERAGHNI, D., AND NECIR, A. (2013). On the asymptotic normality of hill’s estimator of the tail index under random censoring. Preprint: arXiv-1302.1666, 44. CHAOUCH, M. AND KHARDANI, S. (2014). Randomly censored quantile regression estima-tion using functional stationary ergodic data. Journal of Nonparametric Statistics, 27 (1), 65-87. · Zbl 1328.62276 |
| [7] | DE HAAN, L. AND FERREIRA, A. (2007). Extreme value theory: An introduction. Springer Science & Business Media. |
| [8] | DELAFOSSE, E. AND GUILLOU, A. (2002). Almost sure convergence of a tail index estimator in the presence of censoring. Comptes Rendus Mathematique, 335 (4), 375-380. Journal home page: http://www.jafristat.net, www.projecteuclid.org/euclid.as, www.ajol.info/afst · Zbl 1140.60314 |
| [9] | J. U. Rutikanga and Aliou Diop, Afrika Statistika, Vol. 16 (2), 2021, pages 2647 -2688. Functional kernel estimation of the conditional extreme value index under random right censoring. 2667 · Zbl 1485.62060 |
| [10] | DREES, H. (1998). On smooth statistical tail functionals. Scandinavian Journal of Statistics, 25 (1),187-210. · Zbl 0923.62032 |
| [11] | EINMAHL, J. H., FILS-VILLETARD , A. AND GUILLOU, A. (2008). Statistics of extremes under random censoring. Bernoulli, 14 (1), 207-227. · Zbl 1155.62036 |
| [12] | FEI, J., QING, C., GUOSHENG , Y., AND SHEN, H. (2020). Functional cen-sored quantile regression. Journal of the American Statistical Association, 115 (530), 931-944. doi:10.1080/01621459.2019.1602047. URL: https:// doi.org/ 10.1080/01621459.2019.1602047 · Zbl 1445.62336 · doi:10.1080/01621459.2019.1602047 |
| [13] | FERRATY, F. AND VIEU, P. (2004). Nonparametric models for functional data, with appli-cation in regression, time series prediction and curve discrimination. Nonparametric Statistics, 16 (1-2),111125. · Zbl 1049.62039 |
| [14] | FERRATY, F. AND VIEU, P. (2006). Nonparametric Functional Data Analysis: Theory and Practice (Springer Series in Statistics). Springer-Verlag, Berlin, Heidelberg. FERREZ, J., DAVISON, A., AND REBETEZ, M. (2011). Extreme temperature analysis under forest cover compared to an open field. Agricultural and Forest Meteorology, 151 (7), 992-1001. · Zbl 1119.62046 |
| [15] | GARDES, L. AND GIRARD, S. (2008). A moving window approach for nonparametric estima-tion of the conditional tail index. Journal of Multivariate Analysis, 99 (10), 2368-2388. · Zbl 1151.62040 |
| [16] | GARDES, L. AND GIRARD, S. (2010). Conditional extremes from heavy-tailed distributions: An application to the estimation of extreme rainfall return levels. Extremes, 13 (2), 177-204. · Zbl 1238.62136 |
| [17] | GARDES, L. AND GIRARD, S. (2012). Functional kernel estimators of large conditional quan-tiles. Electronic Journal of Statistics, 6, 1715-1744. · Zbl 1295.62052 |
| [18] | GOEGEBEUR, Y., GUILLOU, A., AND OSMANN, M. (2014a). A local moment type estimator for the extreme value index in regression with random covariates.Canadian Journal of Statistics,42(3), 487-507. · Zbl 1297.62116 |
| [19] | GOEGEBEUR,Y., GUILLOU, A. AND SCHORGEN, A. (2014b). Nonparametric regression es-timation of conditional tails: the random covariate case.Statistics,48(4), 732-755. · Zbl 1326.62088 |
| [20] | GOMES, M. I. AND NEVES, M. M. (2010). A note on statistics of extremes for censoring schemes on a heavy right tail.In Information Technology Interfaces (ITI), 2010 32nd In-ternational Conference on. IEEE, 539-544. |
| [21] | GOMES, M.I. AND NEVES, M. M. (2011). Estimation of the extreme value index for randomly censored data.Biometrical Letters,48(1), 1-22. |
| [22] | Lo, G. S. (2018). Mathematical foundations of probability theory. arXiv preprint arXiv:1808.01713 |
| [23] | Ndao P. (2015). Modélisation de valeurs extrêmes conditionnelles en présence de censure. PhD thesis, Université Gaston Berger de Saint-Louis. |
| [24] | NDAO P, DIOP A, DUPUY JF. (2016). Nonparametric estimation of the conditional extreme-value index with random covariates and censoring. Journal of Statistical Planning and Inference, 168:20-37. · Zbl 1333.62138 |
| [25] | PISARENKO, V. AND SORNETTE, D. (2003). Characterization of the frequency of ex-treme earth-quake events by the generalized Pareto distribution.Pure and applied geo-physics,160(12), 2343-2364. |
| [26] | RAMSAY, J. O. AND DALZELL, C. (1991). Some tools for functional data analysis.Journal of the Royal Statistical Society. Series B (Methodological), 539-572. · Zbl 0800.62314 |
| [27] | RAMSAY, J. O. AND SILVERMAN, B. W. (2005). Functional data analysis.Journal of the Royal Statistical Society. Series B (Methodological). · Zbl 1079.62006 |
| [28] | SCHERVISH, M. J. (2012). Theory of statistics. Springer Science & Business Media. |
| [29] | Stupfler G. (2016). Estimating the conditional extreme-value index under random right-censoring. Journal of Multivariate Analysis. 144:1-24. · Zbl 1328.62220 |
| [30] | VAN DERVAART, A. W. (2000).Asymptotic statistics, volume 3. Cambridge university press. Journal home page: http://www.jafristat.net, www.projecteuclid.org/euclid.as, www.ajol.info/afst · Zbl 0943.62002 |
| [31] | J. U. Rutikanga and Aliou Diop, Afrika Statistika, Vol. 16 (2), 2021, pages 2647 -2688. Functional kernel estimation of the conditional extreme value index under random right censoring. 2668 · Zbl 1485.62060 |
| [32] | WORMS, J. AND WORMS, R.(2014). New estimators of the extreme value index under ran-dom right censoring, for heavy-tailed distributions. Extremes, 17(2):337-358. · Zbl 1309.62093 |
| [33] | WORMS, J. AND WORMS, R. (2018). Extreme value statistics for censored data with heavy tails under competing risks. Metrika,81(7), 849-889. · Zbl 1401.62076 |
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