Computer-intensive rate estimation, diverging statistics and scanning. (English) Zbl 1209.62050

McElroy, Tucker; Politis, Dimitris N.

Ann. Stat. 35, No. 4, 1827-1848 (2007).

Summary: A general rate estimation method is proposed that is based on studying the in-sample evolution of appropriately chosen diverging/converging statistics. The proposed rate estimators are based on simple least squares arguments, and are shown to be accurate in a very general setting without requiring the choice of a tuning parameter. The notion of scanning is introduced with the purpose of extracting useful subsamples of the data series; the proposed rate estimation method is applied to different scans, and the resulting estimators are then combined to improve accuracy. Applications to heavy tail index estimation as well as to the problem of estimating the long memory parameter are discussed; a small simulation study complements our theoretical results.

MSC:

62G05	Nonparametric estimation
62G32	Statistics of extreme values; tail inference
65C60	Computational problems in statistics (MSC2010)

Keywords:

convergence rate; heavy tail index; long memory; subsampling

Software:

longmemo

References:

[1]	Barbe, P. and Bertail, P. (1993). Testing the global stability of a linear model. Document de travail of INRA-CORELA, Paris. (See also its revised version, Document de travail CREST, Paris, 2003.)
[2]	Beran, J. (1994). Statistics for Long-Memory Processes . Chapman and Hall, New York. · Zbl 0869.60045
[3]	Bertail, P., Politis, D. N. and Romano, J. P. (1999). On subsampling estimators with unknown rate of convergence. J. Amer. Statist. Assoc. 94 569–579. JSTOR: · Zbl 1072.62551 · doi:10.2307/2670177
[4]	Breiman, L. (1996). Bagging predictors. Machine Learning 24 123–140. · Zbl 0858.68080
[5]	Csörgő, S., Deheuvels, P. and Mason, D. M. (1985). Kernel estimates of the tail index of a distribution. Ann. Statist. 13 1050–1077. · Zbl 0588.62051 · doi:10.1214/aos/1176349656
[6]	Csörgő, S. and Viharos, L. (1998). Estimating the tail index. In Asymptotic Methods in Probability and Statistics (B. Szyszkowicz, ed.) 833–881. North-Holland, Amsterdam. · Zbl 1042.62543
[7]	Danielsson, J., de Haan, L., Peng, L. and de Vries, C. G. (2001). Using a bootstrap method to choose the sample fraction in tail index estimation. J. Multivariate Anal. 76 226–248. · Zbl 0976.62044 · doi:10.1006/jmva.2000.1903
[8]	Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin. · Zbl 0873.62116
[9]	Geweke, J. and Porter-Hudak, S. (1983). The estimation and application of long memory time series models. J. Time Ser. Anal. 4 221–238. · Zbl 0534.62062 · doi:10.1111/j.1467-9892.1983.tb00371.x
[10]	Giraitis, L., Robinson, P. M. and Surgailis, D. (1999). Variance-type estimation of long memory. Stochastic Process. Appl. 80 1–24. · Zbl 0955.62090 · doi:10.1016/S0304-4149(98)00062-3
[11]	Gomes, M. I., de Haan, L. and Pestana, D. (2004). Joint exceedances of the ARCH process. J. Appl. Probab. 41 919–926. · Zbl 1065.60054 · doi:10.1239/jap/1091543434
[12]	Hurst, H. E. (1951). Long-term storage capacity of reservoirs. Trans. Amer. Soc. Civil Engineers 116 770–808.
[13]	Ibragimov, I. A. and Rozanov, Y. A. (1978). Gaussian Random Processes . Springer, New York. · Zbl 0392.60037
[14]	McElroy, T. and Politis, D. N. (2002). Robust inference for the mean in the presence of serial correlation and heavy-tailed distributions. Econometric Theory 18 1019–1039. JSTOR: · Zbl 1023.62088 · doi:10.1017/S026646660218501X
[15]	McElroy, T. and Politis, D. N. (2007). Self-normalization for heavy-tailed time series with long memory. Statist. Sinica 17 199–220. · Zbl 1145.62073
[16]	Meerschaert, M. and Scheffler, H. (1998). A simple robust estimation method for the thickness of heavy tails. J. Statist. Plann. Inference 71 19–34. · Zbl 0931.62022 · doi:10.1016/S0378-3758(98)00093-7
[17]	Politis, D. N. (2002). A new approach on estimation of the tail index. C. R. Math. Acad. Sci. Paris 335 279–282. · Zbl 1004.62052 · doi:10.1016/S1631-073X(02)02450-0
[18]	Politis, D. N. and Romano, J. P. (1994). Large sample confidence regions based on subsamples under minimal assumptions. Ann. Statist. 22 2031–2050. · Zbl 0828.62044 · doi:10.1214/aos/1176325770
[19]	Politis, D. N., Romano, J. P. and Wolf, M. (1999). Subsampling. Springer, New York. · Zbl 0931.62035
[20]	Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrsch. Verw. Gebiete 31 287–302. · Zbl 0303.60033 · doi:10.1007/BF00532868
[21]	Taqqu, M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrsch. Verw. Gebiete 50 53–83. · Zbl 0397.60028 · doi:10.1007/BF00535674