Flexible extreme value inference. (English) Zbl 1536.62057
Summary: We introduce flexible small sample modeling for extremes by introducing the new numerical characteristics of heavy tail. We illustrate in this article advantages of such flexibility. In this article, we show that we can obtain asymptotic normality of generalized Hill estimators by application of Karamata’s representation for regularly varying tails. Second order regularity conditions however better relates to Edgeworth types of normal approximations albeit requiring larger data samples. Finally both expansions are prone for bootstrap and other subsampling techniques. All existing results indicate that proper representation of tail behavior play a special and somewhat intriguing role in that context. The application of this new methodology is simple and flexible, handsome for real data sets. Alternative and powerful versions of the Hill plot are also introduced and illustrated on real data of snow extremes from Slovakia. We also demonstrate the importance of box-plot based techniques for small samples.
MSC:
| 62G32 | Statistics of extreme values; tail inference |
| 62G30 | Order statistics; empirical distribution functions |
Keywords:
asymptotic normality; Hill estimator; t-Hill estimator; harmonic mean estimator; Hill-plot; Karamata representationReferences:
| [1] | de Haan, L., Stadtmüller, U. (1996). Generalized regular variation of second order. J. Aust. Math. Soc. A. 61(3):381-395. DOI: . · Zbl 0878.26002 |
| [2] | de Haan, L., Ferreira, A. (2006). Extreme Value Theory: An Introduction, Springer Series in Operations Research and Financial Engineering. New York: Springer. · Zbl 1101.62002 |
| [3] | Geluk, J., de Haan, L., Resnick, S., Stărică, C. (1997). Second-order regular variation, convolution and the central limit theorem. Stochastic Process. Appl. 69(2):139-159. DOI: . · Zbl 0913.60001 |
| [4] | Hall, P. (1978). Representations and limit theorems for extreme value distributions. J. Appl. Probab.15(3):639-644. DOI: . · Zbl 0388.60030 |
| [5] | Hall, P. (1982). On some simple estimates of an exponent of regular variation. J. R. Stat. Soc. Ser. B. 44(1):37-42. DOI: . · Zbl 0521.62024 |
| [6] | Lo, G.S., Fall, A. M. (2011). Another look at second order condition in extreme value theory. AFST. 6(1):346-370. DOI: . · Zbl 1258.62058 |
| [7] | Ivette Gomes, M., Pestana, D. (2007). A sturdy reduced-bias extreme quantile (VaR) estimator. J. Am. Stat. Assoc.102(477):280-292. DOI: . · Zbl 1284.62300 |
| [8] | Cheng, S., Pan, J. (1998). Asymptotic expansions of estimators for the tail index with applications. Scand. J. Statist. 25(4):717-728. DOI: . · Zbl 0927.62046 |
| [9] | Beran, J., Schell, D., Stehlík, M. (2014). The harmonic moment tail index estimator: Asymptotic distribution and robustness. Ann. Inst. Stat. Math. 66(1):193-220. DOI: . · Zbl 1281.62123 |
| [10] | Hill, B. (1975). A simple general approach to inference about the tail of a distribution. Ann. Statist. 3(5):1163-1174. DOI: . · Zbl 0323.62033 |
| [11] | Brilhante, M.F., Gomes, M.I., Pestana, D. (2013). A simple generalization of the Hill estimator. Comput. Stat. Data Anal. 57(1):518-535. DOI: . · Zbl 1365.62148 |
| [12] | Caeiro, F., Gomes, M. I., Beirlant, J., de Wet, T. (2016). Mean-of-order p reduced-bias extreme value index estimation under a third-order framework. Extremes. 19(4):561-589. DOI: . · Zbl 1357.62211 |
| [13] | Karamata, J. (1930). Sur un mode de croissance réguliére des fonctions. Mathematica (Cluj).4:38-53. · JFM 56.0907.01 |
| [14] | Resnick, S.I. (1987). Extreme Values, Regular Variation and Point Processes. New York: Springer. · Zbl 0633.60001 |
| [15] | Karamata, J. (1963). Some Theorems Concerning Slowly Varying Functions, Vol. 432. Madison, Wisconsin: Math. Res. Center, U.S. Army, Tech. Sum. Report. |
| [16] | Vuilleumier, M. (1976). Slowly varying functions in the complex plane. Trans. Am. Math. Soc. 218(0):343-348. DOI: . · Zbl 0351.30012 |
| [17] | Buldygin, V. V., Klesov, O. I., Steinebach, J. G. (2006). On some extensions of Karamata’s theory and their applications. Publ. Inst. Math. (Belgr). 80(94):59-96. DOI: . · Zbl 1199.26001 |
| [18] | Drees, H., de Haan, L.A., Resnick, S.I. (2000). How to make a hill plot. Ann. Statist. 28(1):254-274. DOI: . · Zbl 1106.62333 |
| [19] | Davis, R., Resnick, S.I. (1984). Tail estimates motivated by extreme value theory. Ann. Statist. 12(4):1467-1487. DOI: . · Zbl 0555.62035 |
| [20] | Stehlík, M. (2003). Distributions of exact tests in the exponential family. Metrika.57:145-164. · Zbl 1433.62066 |
| [21] | Haeusler, E., Teugels, J. L. (1985). On asymptotic normality of Hill’s estimator for the exponent of regular variation. Ann. Statist. 13(2):743-756. DOI: . · Zbl 0606.62019 |
| [22] | Pancheva, E., Jordanova, P. (2012). Weak asymptotic results for t-Hill estimator. Comptes Rend. Acad. Bulg. Sci.65(12):1649-1656. · Zbl 1313.62026 |
| [23] | Embrechts, P., Klüppelberg, C., Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Berlin-Heidelberg: Springer. · Zbl 0873.62116 |
| [24] | Paulauskas, V., Vaiciulis, M. (2013). On the improvement of Hill and some others estimators. Lith. Math. J. 53(3):336-355. DOI: . · Zbl 1294.62110 |
| [25] | Pickands, J. III (1975). Statistical inference using extreme order statistics. Ann. Statist. 3(1):119-131. [R] DOI: . · Zbl 0312.62038 |
| [26] | StehlíK, M., Sadovský, Z., Jordanov, P. (2015). Statistical analysis related to exceptional snow loads. Appl. Math. Inf. Sci. 9(1L):19-27. DOI: . |
| [27] | de Haan, L. (1970). On Regular Variation and Its Application to the Weak Convergence of Sample Extremes. Mathematical Centre Tract, Vol. 32, Amsterdam, Holland: Mathematics Centre. · Zbl 0226.60039 |
| [28] | Mason, D. M. (1982). Laws of Large Numbers for Sums of Extreme Values. Ann. Probab. 10(3):754-764. DOI: . · Zbl 0493.60039 |
| [29] | Resnick, S.I. (2006). Heavy-Tailed Phenomena, Probabilistic and Statistical Modeling. Springer. |
| [30] | Hall, P. (1927). The distribution of means for samples of size n drawn from a population in which the variate takes values between 0 and 1, All such values being equally probable. Biometrika.19(3/4):240-245. DOI: . · JFM 53.0518.05 |
| [31] | Irwin, J.O. (1927). On the frequency distribution of the means of samples from a population having any law of frequency with finite moments, with special reference to Pearson’s Type II. Biometrika.19(3-4):225-239. DOI: . · JFM 53.0518.04 |
| [32] | Billingsley, P. (1977). Convergence of Probability Measures, 2nd erd, Canada: Wiley. |
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.
