Bias-reduced extreme quantile estimators of Weibull tail-distributions. (English) Zbl 1250.62024
Summary: We consider the problem of estimating an extreme quantile of a Weibull tail-distribution. The new extreme quantile estimator has a reduced bias compared to the more classical ones proposed in the literature. It is based on an exponential regression model that was introduced by the authors [Test 17, No. 2, 311–331 (2008; Zbl 1196.62052)]. The asymptotic normality of the extreme quantile estimator is established. We also introduce an adaptive selection procedure to determine the number of upper order statistics to be used. A simulation study as well as an application to a real data set is provided in order to prove the efficiency of the above-mentioned methods.
MSC:
| 62G32 | Statistics of extreme values; tail inference |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62G30 | Order statistics; empirical distribution functions |
| 62G05 | Nonparametric estimation |
| 65C60 | Computational problems in statistics (MSC2010) |
Citations:
Zbl 1196.62052References:
| [1] | Beirlant, J.; Broniatowski, M.; Teugels, J. L.; Vynckier, P., The mean residual life function at great age: applications to tail estimation, J. Statist. Plann. Inference, 45, 21-48 (1995) · Zbl 0846.62025 |
| [2] | Beirlant, J.; Teugels, J.; Vynckier, P., Practical Analysis of Extreme Values (1996), Leuven University Press: Leuven University Press Leuven · Zbl 0888.62003 |
| [3] | Beirlant, J.; Dierckx, G.; Goegebeur, Y.; Matthys, G., Tail index estimation and an exponential regression model, Extremes, 2, 177-200 (1999) · Zbl 0947.62034 |
| [4] | Beirlant, J.; Dierckx, G.; Guillou, A.; Starica, C., On exponential representations of log-spacings of extreme order statistics, Extremes, 5, 157-180 (2002) · Zbl 1036.62040 |
| [5] | Beirlant, J.; Bouquiaux, C.; Werker, B., Semiparametric lower bounds for tail index estimation, J. Statist. Plann. Inference, 136, 705-729 (2006) · Zbl 1077.62039 |
| [6] | Breiman, L.; Stone, C. J.; Kooperberg, C., Robust confidence bounds for extreme upper quantiles, J. Comput. Statist. Simulation, 37, 127-149 (1990) · Zbl 0775.62117 |
| [7] | Broniatowski, M., On the estimation of the Weibull tail coefficient, J. Statist. Plann. Inference, 35, 349-366 (1993) · Zbl 0771.62026 |
| [8] | Davison, A.; Smith, R., Models for exceedances over high thresholds, J. Roy. Statist. Soc. Ser. B, 52, 393-442 (1990) · Zbl 0706.62039 |
| [9] | Diebolt, J.; El-Aroui, M.; Garrido, M.; Girard, S., Quasi-conjugate Bayes estimates for GPD parameters and application to heavy tails modelling, Extremes, 8, 57-78 (2005) · Zbl 1091.62009 |
| [10] | Diebolt, J., Gardes, L., Girard, S., Guillou, A., 2007. Bias-reduced estimators of the Weibull-tail coefficient. Test, to appear.; Diebolt, J., Gardes, L., Girard, S., Guillou, A., 2007. Bias-reduced estimators of the Weibull-tail coefficient. Test, to appear. · Zbl 1196.62052 |
| [11] | Ditlevsen, O., Distribution arbitrariness in structural reliability, (Structural Safety and Reliability (1994), Balkema: Balkema Rotterdam), 1241-1247 |
| [12] | Drees, H., On smooth statistical tail functionals, Scand. J. Statist., 25, 187-210 (1998) · Zbl 0923.62032 |
| [13] | Embrechts, P.; Klüppelberg, C.; Mikosch, T., Modelling Extremal Events (1997), Springer: Springer Berlin · Zbl 0873.62116 |
| [14] | Feuerverger, A.; Hall, P., Estimating a tail exponent by modelling departure from a pareto distribution, Ann. Statist., 27, 760-781 (1999) · Zbl 0942.62059 |
| [15] | Gardes, L.; Girard, S., Estimating extreme quantiles of Weibull tail-distributions, Comm. Statist. Theory Methods, 34, 1065-1080 (2005) · Zbl 1073.62046 |
| [16] | Gardes, L.; Girard, S., Comparison of Weibull tail-coefficient estimators, REVSTAT Statist. J., 4, 163-188 (2006) · Zbl 1141.62341 |
| [17] | Geluk, J.L., de Haan, L., 1987. Regular Variation, Extensions and Tauberian Theorems. In: Math Centre Tracts, vol. 40. Centre for Mathematics and Computer Science, Amsterdam.; Geluk, J.L., de Haan, L., 1987. Regular Variation, Extensions and Tauberian Theorems. In: Math Centre Tracts, vol. 40. Centre for Mathematics and Computer Science, Amsterdam. · Zbl 0624.26003 |
| [18] | Girard, S., A Hill type estimate of the Weibull tail-coefficient, Comm. Statist. Theory Methods, 33, 205-234 (2004) · Zbl 1066.62052 |
| [19] | de Haan, L.; Rootzén, H., On the estimation of high quantiles, J. Statist. Plann. Inference, 35, 1-13 (1993) · Zbl 0770.62026 |
| [20] | Hall, P.; Welsh, A. H., Adaptive estimates of parameters of regular variation, Ann. Statist., 13, 331-341 (1985) · Zbl 0605.62033 |
| [21] | Hosking, J.; Wallis, J., Parameter and quantile estimation for the generalized Pareto distribution, Technometrics, 29, 339-349 (1987) · Zbl 0628.62019 |
| [22] | Matthys, G.; Delafosse, E.; Guillou, A.; Beirlant, J., Estimating catastrophic quantile levels for heavy-tailed distributions, Insurance Math. Econom., 34, 517-537 (2004) · Zbl 1188.91237 |
| [23] | Smith, J., Estimating the upper tail of flood frequency distributions, Water Resources Res., 23, 1657-1666 (1991) |
| [24] | Weissman, I., Estimation of parameters and large quantiles based on the \(k\) largest observations, J. Amer. Statist. Assoc., 73, 812-815 (1978) · Zbl 0397.62034 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
