## Estimation of the extreme-value index and generalized quantile plots.(English)Zbl 1123.62034

Summary: In extreme-value analysis, a central topic is the adaptive estimation of the extreme-value index $$\gamma$$. Hitherto, most of the attention in this area has been devoted to the case $$\gamma>0$$, that is, when $$\overline{F}$$ is a regularly varying function with index $$-1/\gamma$$. In addition to the well-known B. M. Hill estimator [Ann. Stat. 3, 1163–1174 (1975; Zbl 0323.62033)], many other estimators are currently available. Among the most important are the kernel-type estimators and the weighted least-squares slope estimators based on the Pareto quantile plot or the Zipf plot, as reviewed by S. Csörgő and L. Viharos [Estimating the tail index. B. Szyszkowicz (ed.), Asymptotic Methods Probab. Stat., 833–881 (1998; Zbl 1042.62543)]. Using an exponential regression model (ERM) for spacings between successive extreme order statistics, both J. Beirlant et al. [Extremes 2, No. 2, 177–200 (!999; Zbl 0947.62034)] and A. Feuerverger and P. Hall [Ann. Stat. 27, No. 2, 760–781 (1999: Zbl 0942.62059)] introduced bias-reduced estimators.
For the case where $$\gamma$$ is real, Hill’s estimator has been generalized to a moment-type estimator by A. L. M. Dekkers et al. [ibid. 17, No. 4, 1833–1855 (1989; Zbl 0701.62029)]. Alternatively, J. Beirlant et al. [J. Am. Stat. Assoc. 91, No. 436, 1659–1667 (1996; Zbl 0881.62077)] introduced a Hill-type estimator that is based on the generalized quantile plot. Another popular estimation method follows from maximum likelihood estimation applied to the generalizations of the Pareto distribution. In the present paper, slope estimators for $$\gamma>0$$ are generalized to the case where $$\gamma$$ is real-valued. This is accomplished by replacing the Zipf plot by a generalized quantile plot. We make an asymptotic comparison of our estimator with the moment estimator and with the maximum likelihood estimator. A case study illustrates our findings. Finally, we offer a regression model that generalizes the ERM in that it allows the construction of bias-reduced estimators. Moreover, the model provides an adaptive selection rule for the number of extremes needed in several of the existing estimators.

### MSC:

 62G32 Statistics of extreme values; tail inference 62G20 Asymptotic properties of nonparametric inference 62G05 Nonparametric estimation 62G08 Nonparametric regression and quantile regression
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### References:

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