Abstract
As a leading statistician in extreme value theory, Professor Laurens de Haan has made significant contribution in both probability and statistics of extremes. In honor of his 70th birthday, we review testing issues in extremes, which include research done by Professor Laurens de Haan and many others. In comparison with statistical estimation in extremes, research on testing has received less attention. So we also point out some practical questions in this direction.
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References
Arnold, B.C., Balakrishnan, N.: Relations, Bounds, and Approximations for Order Statistics. Springer (1989)
Balakrishnan, N., Cohen, A.: Order Statistics and Inference: Estimation Methods. Academic Press (1990)
Beirlant, J., de Wet, T., Goegebeur, Y.: A goodness-of-fit statistic for Pareto-type behaviour. J. Comput. Appl. Math. 186, 99–116 (2006)
Beirlant, J., Goegebeur, Y.: Local polynomial maximum likelihood estimation for Pareto-type distributions. J. Multi. Anal. 89, 97–118 (2004)
Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes: Theory and Applications. Wiley (2004)
Beisel, C.J., Rokyta, D.R., Wichman, H.A., Joyce, P.: Testing the extreme value domain of attraction for distributions of beneficial fitness effects. Genetics 176, 2441–2449 (2007)
Berk, R.H., Jones, D.H.: Goodness-of-fit-statistics that dominates the Kolmogorov statistics. Z. Wahrscheinlichkeitstheorie Verw. Geb. 47, 47–59 (1979)
Berman, S.: Sojourns and Extremes of Stochastic Processes. Chapman & Hall/CRC (1992)
Cabana, A., Quiroz, A.J.: Using the empirical moment generating function in testing for the Weibull and the type I extreme value distributions. Test 14, 417–431 (2005)
Cao, R., Van Keilegom, I.: Empirical likelihood tests for two-sample problems via nonparametric density estimation. Canad. J. Statist. 34, 61–77 (2006)
Castillo, E.: Extreme Value Theory in Engineering. Academic Press. (1988)
Castillo, E., Hadi, A.S.: Fitting the generalized Pareto distribution to data. J. Amer. Statist. Assoc. 92(440), 1609–1620 (1997)
Castillo, E., Hadi, A.S., Balakrishnan, N., Sarabia, J.M.: Extreme Value and Related Models with Applications in Engineering and Science. Wiley-Interscience (2004)
Chavez-Demoulin, V., Davison, A.C.: Generalized additive modelling of sample extremes. J. Roy. Statist. Soc. Ser. C 54, 207–222 (2005)
Chavez-Demoulin, V., Embrechts, P.: Smooth extremal models in finance. J. Risk Insurance 71, 183–199 (2004)
Chen, S.X., Gao, J.: An adaptive empirical likelihood test for time series models. J. Econom. 141, 950–972 (2007)
Chen, S.X., Härdle, W., Li, M.: An empirical likelihood goodness-of-fit test for time series. J. Roy. Statist. Soc. Ser. B 65, 663–678 (2003)
Chen, S.X., Van Keilegom, I.: A goodness-of-fit test for parametric and semiparametric models in multiresponse regression. Technical report (2006)
Cheng, M., Peng, L.: Variance reduction in multivariate likelihood models. J. Amer. Statist. Assoc. 102(477), 293–304 (2007)
Choulakian, V., Stephens, M.A.: Goodness-of-fit tests for the generalized Pareto distribution. Technometrics 43(4), 478–484 (2001)
Coles, S.: An Introduction to Statistical Modeling of Extreme Values. Springer (2001)
Coles, S.G., Tawn, J.A.: Statistics of coastal flood prevention. Phil. Trans. Royal Soc. London A 332, 457–476 (1990)
David, H.A., Nagaraja, H.N.: Order Statistics. Wiley-Interscience, 3rd edn. (2003)
Davison, A.C., Ramesh, N.I.: Local likelihood smoothing of sample extremes. J. Roy. Statist. Soc. Ser. B 62, 191–208 (2000)
Davison, A.C., Smith, R.L.: Models for exceedances over high thresholds (with discussion). J. Roy. Statist. Soc. Ser. B 52, 393–442 (1990)
de Haan, L., Ferreira, A.: Extreme Value Theory: An Introduction. Springer (2006)
de Haan, L., Peng, L., Neves, C.: Parametric tail copula estimation and model testing. Technical report (2007)
Deheuvels, P., Martynov, G.V.: Cramer-von Mises-type tests with applications to tests of independence for multivariate extreme-value distributions. Comm. Statist. Theory Methods 25(4), 871–908 (1996)
Dietrich, D., de Haan, L., Hüsler, J.: Testing extreme value conditions. Extremes 5, 71–85 (2002)
Dixon, M.J., Tawn, J.A.: The effect of non-stationarity on extreme sea-level estimation. Appl. Statist. 48, 135–151 (1999)
Draisma, G., Drees, H., Ferreira, A., de Haan, L.: Bivariate tail estimation: dependence in asymptotic independence. Bernoulli 10, 251–280 (2004)
Drees, H., Ferreira, A., de Haan, L.: On maximum likelihood estimation of the extreme value index. Ann. Appl. Prob. 14, 1179–1201 (2004)
Drees, H., de Haan, L., Li, D.: Approximations to the tail empirical distribution function with application to testing extreme value conditions. J. Statist. Plann. Inf. 136, 3498–3538 (2006)
Einmahl, J.H.J., de Haan, L., Li, D.: Weighted approximations of tail copula processes with application to testing the bivariate extreme value condition. Ann. Statist. 34, 1987–2014 (2006)
Einmahl, J.H.J., McKeague, I.W.: Empirical likelihood based hypothesis testing. Bernoulli 9, 267–290 (2003)
Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Springer (1997)
Falk, M.: On testing the extreme value index via the POT-method. Ann. Statist. 23, 2013–2035 (1995)
Falk, M., Hüsler, J., Reiss, R.D.: Laws of Small Numbers: Extremes and Rare Events. Birkhauser Basel, 2nd edn. (2005)
Falk, M., Michel, R.: Testing for tail independence in extreme value models. Ann. Inst. Statist. Math. 58, 261–290 (2006)
Finkenstadt, B., Rootzen, H.: Extreme Values in Finance, Telecommunication and the Environment. Chapman & Hall (2003)
Fraga Alves, M.I., Gomes, M.I.: Statistical choice of extreme value domain of attraction—a comparative analysis. Comm. Statist.- Theory Methods 25, 789–811 (1996)
Fraga Alves, M.I., Neves, C.: Testing extreme value conditions – an overview and recent approaches. Technical report (2006)
Galambos, J.: The Asymptotic Theory of Extreme Order Statistics. Krieger Pub. Co., 2nd ed. (1987)
Galambos, J., Lechner, J., Simiu, E.: Extreme Value Theory and Applications. Springer (1994)
Gijbels, I., Peng, L.: Estimation of a support curve via order statistics. Extremes 3, 251–277 (2000)
Gomes, M.I.: Generalized Gumbel and likelihood ratio test statistics in the multivariate GEV model. Comput. Stat. Data Anal. 7, 259–267 (1989)
Gomes, M.I., Teresa, A.M.: Inference in a multivariate generalized extreme value model—asymptotic properties of two test statistics. Scand. J. Statist. 13, 291–300 (1986)
Gumbel, E.J.: Statistics of Extremes. Columbia University Press, New York (1958)
Hall, P., Nussbaum, M., Stern, S.E.: On the estimation of a support of indeterminate sharpness. J. Multi. Anal. 62, 304–232 (1997)
Hall, P., Tajvidi, N.: Nonparametric analysis of temporal trend when fitting parametric models to extreme-value data. Statist. Sci. 15, 153–167 (2000)
Hall, P., Van Keilegom, I.: Nonparametric “regression” when errors are centred at endpoints. Technical report (2006)
Hasofer, A.M., Li, S.: Estimation for type II domain of attraction based on the W statistic. Austral. & New Zealand J. Statist. 41(2), 223–232 (1999)
Hasofer, A.M., Wang, Z.: A test for extreme value domain of attraction. J. Amer. Statist. Assoc. 87(417), 171–177 (1992)
Hassanein, K.M., Saleh, A.K.: Testing equality of locations and quantiles of several extreme-value distributions by use of few order statistics of samples from extreme-value and Weibull distributions. In: Order Statistics and Nonparametrics: Theory and Applications, pp. 115–132. North-Holland, Amsterdam (1992)
Hassanein, K.M., Saleh, A.K., Brown, E.F.: Estimation and testing of quantiles of the extreme-value distribution. J. Statist. Plann. Inf. 14, 389–400 (1986)
Hosking, J.R.M.: Testing whether the shape parameter is zero in the generalized extreme-value distribution. Biometrika 71, 367–374 (1984)
Hüsler, J., Li, D.: On testing extreme value conditions. Extremes 9, 69–86 (2006)
Hüsler, J., Li, D.: Testing extreme value conditions with applications. In Statistical Analysis of Extreme Values: with Applications to Insurance, Finance, Hydrology and Other Fields. R.-D. Reiss, Thomas, M. (2007). Birkhäuser Boston. 3rd ed. pp. 144–151 (2007a)
Hüsler, J., Li, D.: Testing asymptotic independence in bivariate extremes. Technical report (2007b)
Janic-Wroblewska, A.: Data-driven smooth tests for the extreme value distribution. Statistics 38, 413–426 (2004)
Jureckova, J.: Test of tails based on extreme regression quantiles. Stat. Probab. Lett. 49, 53–61 (2000)
Jureckova, J.: Statistical tests on tail index of a probability distribution. Metron 61, 151–175 (2003)
Jureckova, J., Koul, H.L., Picek, J.: Testing the tail index in autoregressive models. Ann. Inst. Statist. Math. (2007) (to appear)
Jureckova, J., Picek, J.: A class of tests on the tail index. Extremes 4, 165–183 (2001)
Koning, A., Peng, L.: Goodness-of-fit tests for a heavy tailed distribution. Technical report (2007)
Kotz, S., Nadarajah, S.: Extreme Value Distributions: Theory and Applications. World Scientific Publishing Company (2001)
Lawless, J.F.: Confidence interval estimation for the Weibull and extreme value distributions. Technometrics 20, 355–365 (1978)
Lawless, J.F., Mann, N.R.: Tests for homogeneity of extreme value scale parameters. Comm. Statist.—Theory Methods A 5, 389–405 (1976)
Leadbetter, M.R., Lindgren, G., Rootzen, H.: Extremes and related properties of stationary sequences and processes. Springer, New York (1983)
Ledford, A.W., Tawn, J.A.: Statistics for near independence in multivariate extreme values. Biometrika 83, 169–187 (1996)
Ledford, A.W., Tawn, J.A.: Diagnostics for dependence within time series extremes. J. R. Statist. Soc. B 65, 521–543 (2003)
Li, G., Van Keilegom, I.: Likelihood ratio confidence bands in nonparametric regression with censored data. Scand. J. Statist. 29, 547–562 (2002)
Liao, M., Shimokawa, T.: A new goodness-of-fit test for type-I extreme-value and 2-parameter Weibull distributions with estimated parameters. J. Statist. Comput. Simul. 64, 23–48 (1999)
Lockhart, R.A., O’Reilly, F., Stephens, M.A.: Tests for the extreme value and Weibull distributions based on normalized spacings. Naval Res. Logist. Quart. 33, 413–421 (1986)
Mann, N.R., Scheuer, E.M., Fertig, K.W.: A new goodness-of-fit test for the two-parameter Weibull or extreme-value distribution with unknown parameters. Comm. Statist. 2, 383–400 (1973)
Marohn, F.: Testing the Gumbel hypothesis via the POT-method. Extremes 1, 191–213 (1998a)
Marohn, F.: An adaptive efficient test for Gumbel domain of attraction. Scand. J. Statist. 25, 311–324 (1998b)
Marohn, F.: Testing extreme value models. Extremes 3, 363–384 (2000)
Marohn, F.: A characterization of generalized Pareto distributions by progressive censoring schemes and goodness-of-fit tests. Commun. Statist.—Theory Meth. 31(7), 1055–1065 (2002)
McCormick, W.P., Sun, J.: Extreme Value Theory with S Programming. Chapman & Hall/CRC (2008)
Neves, C., Fraga Alves, M.I.: Semi-parametric approach to Hasofer-Wang and Greenwood statistics in extremes. Test 16, 297–313 (2007)
Neves, C., Picek, J., Fraga Alves, M.I.: The contribution of the maximum to the sum of excesses for testing max-domains of attraction. J. Statist. Plann. Inf. 136, 1281–1301 (2006)
Owen, A.B.: Empirical Likelihood. Chapman & Hall/CRC (2001)
Öztürk, A.: On the W test for the extreme value distribution. Biometrika 73, 738–740 (1986)
Öztürk, A., Korukoglu, S.: A new test for the extreme value distribution. Comm. Statist. Simulation Comput. 17, 1375–1393 (1988)
Peng, L.: Estimation of the coefficient of tail dependence in bivariate extremes. Stat. Probab. Lett. 43, 399–409 (1999)
Peng, L.: Bias-corrected estimators for monotone and concave frontier functions. J. Statist. Plann. Inf. 119, 263–275 (2004)
Ramesh, N.I., Davison, A.C.: Local models for exploratory analysis of hydrological extremes. J. Hydrology 256, 106–119 (2002)
Ramos, A., Ledford, A.: Regular score tests of independence in multivariate extreme values. Extremes 8, 5–26 (2005)
Reiss, R.D.: Approximate Distributions of Order Statistics. Springer (1989)
Reiss, R.D., Thomas, M.: Statistical Analysis of Extreme Values: with Applications to Insurance, Finance, Hydrology and Other Fields. Birkhauser Boston. 3rd edn. (2007)
Resnick, S.I.: Extreme Values, Regular Variation and Point Processes. Springer-Verlag (1987)
Salvadori, G., de Michele, C., Kottegoda, N.T., Rosso, R.: Extremes in Nature: An Approach Using Copulas. Springer (2007)
Segers, J., Teugels, J.: Testing the Gumbel hypothesis by Galton’s ratio. Extremes 3, 291–303 (2000)
Shi, D.J.: An analysis of variance test for the extreme value distribution. J. Tianjin Univ. 2, 116–121 (1988)
Sivakumar, M.V.K., Motha, R.P., Das, H.P.: Natural Disasters and Extreme Events in Agriculture: Impacts and Mitigation. Springer (2005)
Stephens, M.A.: Goodness of fit for the extreme value distribution. Biometrika 64, 583–588 (1977)
Tiago de Oliveira, J.: Statistical Extremes and Applications. Springer (1984)
Tiago de Oliveira, J., Gomes, M.I.: Two test statistics for choice of univariate extreme models. In: Tiago de Oliveira, J. (ed.) Statistical Extremes and Applications, D. Reidel, Dordrecht, pp. 651–668 (1984)
Wang, J.Z., Cooke, P., Li, S.: Determination of domains of attraction based on a sequence of maxima. Austral. J. Statist. 38, 173–181 (1996)
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Hüsler, J., Peng, L. Review of testing issues in extremes: in honor of Professor Laurens de Haan. Extremes 11, 99–111 (2008). https://doi.org/10.1007/s10687-007-0052-0
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DOI: https://doi.org/10.1007/s10687-007-0052-0