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Review of testing issues in extremes: in honor of Professor Laurens de Haan

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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)

    Article  MATH  MathSciNet  Google Scholar 

  • Beirlant, J., Goegebeur, Y.: Local polynomial maximum likelihood estimation for Pareto-type distributions. J. Multi. Anal. 89, 97–118 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  Google Scholar 

  • Berk, R.H., Jones, D.H.: Goodness-of-fit-statistics that dominates the Kolmogorov statistics. Z. Wahrscheinlichkeitstheorie Verw. Geb. 47, 47–59 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • Cao, R., Van Keilegom, I.: Empirical likelihood tests for two-sample problems via nonparametric density estimation. Canad. J. Statist. 34, 61–77 (2006)

    MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • Chavez-Demoulin, V., Embrechts, P.: Smooth extremal models in finance. J. Risk Insurance 71, 183–199 (2004)

    Article  Google Scholar 

  • Chen, S.X., Gao, J.: An adaptive empirical likelihood test for time series models. J. Econom. 141, 950–972 (2007)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • Choulakian, V., Stephens, M.A.: Goodness-of-fit tests for the generalized Pareto distribution. Technometrics 43(4), 478–484 (2001)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • Davison, A.C., Smith, R.L.: Models for exceedances over high thresholds (with discussion). J. Roy. Statist. Soc. Ser. B 52, 393–442 (1990)

    MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • Dietrich, D., de Haan, L., Hüsler, J.: Testing extreme value conditions. Extremes 5, 71–85 (2002)

    Article  MathSciNet  Google Scholar 

  • Dixon, M.J., Tawn, J.A.: The effect of non-stationarity on extreme sea-level estimation. Appl. Statist. 48, 135–151 (1999)

    MATH  Google Scholar 

  • Draisma, G., Drees, H., Ferreira, A., de Haan, L.: Bivariate tail estimation: dependence in asymptotic independence. Bernoulli 10, 251–280 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Drees, H., Ferreira, A., de Haan, L.: On maximum likelihood estimation of the extreme value index. Ann. Appl. Prob. 14, 1179–1201 (2004)

    Article  MATH  Google Scholar 

  • 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)

    Article  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  Google Scholar 

  • Einmahl, J.H.J., McKeague, I.W.: Empirical likelihood based hypothesis testing. Bernoulli 9, 267–290 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • Gomes, M.I.: Generalized Gumbel and likelihood ratio test statistics in the multivariate GEV model. Comput. Stat. Data Anal. 7, 259–267 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    MathSciNet  MATH  Google Scholar 

  • Gumbel, E.J.: Statistics of Extremes. Columbia University Press, New York (1958)

    MATH  Google Scholar 

  • Hall, P., Nussbaum, M., Stern, S.E.: On the estimation of a support of indeterminate sharpness. J. Multi. Anal. 62, 304–232 (1997)

    Article  MathSciNet  Google Scholar 

  • Hall, P., Tajvidi, N.: Nonparametric analysis of temporal trend when fitting parametric models to extreme-value data. Statist. Sci. 15, 153–167 (2000)

    Article  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • Hasofer, A.M., Wang, Z.: A test for extreme value domain of attraction. J. Amer. Statist. Assoc. 87(417), 171–177 (1992)

    Article  Google Scholar 

  • 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)

    Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • Hosking, J.R.M.: Testing whether the shape parameter is zero in the generalized extreme-value distribution. Biometrika 71, 367–374 (1984)

    MathSciNet  Google Scholar 

  • Hüsler, J., Li, D.: On testing extreme value conditions. Extremes 9, 69–86 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    MATH  MathSciNet  Google Scholar 

  • Jureckova, J.: Test of tails based on extreme regression quantiles. Stat. Probab. Lett. 49, 53–61 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Jureckova, J.: Statistical tests on tail index of a probability distribution. Metron 61, 151–175 (2003)

    MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • Lawless, J.F., Mann, N.R.: Tests for homogeneity of extreme value scale parameters. Comm. Statist.—Theory Methods A 5, 389–405 (1976)

    Article  MathSciNet  Google Scholar 

  • Leadbetter, M.R., Lindgren, G., Rootzen, H.: Extremes and related properties of stationary sequences and processes. Springer, New York (1983)

    Google Scholar 

  • Ledford, A.W., Tawn, J.A.: Statistics for near independence in multivariate extreme values. Biometrika 83, 169–187 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  • Ledford, A.W., Tawn, J.A.: Diagnostics for dependence within time series extremes. J. R. Statist. Soc. B 65, 521–543 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Li, G., Van Keilegom, I.: Likelihood ratio confidence bands in nonparametric regression with censored data. Scand. J. Statist. 29, 547–562 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    MathSciNet  Google Scholar 

  • Marohn, F.: Testing the Gumbel hypothesis via the POT-method. Extremes 1, 191–213 (1998a)

    Article  MATH  MathSciNet  Google Scholar 

  • Marohn, F.: An adaptive efficient test for Gumbel domain of attraction. Scand. J. Statist. 25, 311–324 (1998b)

    Article  MATH  MathSciNet  Google Scholar 

  • Marohn, F.: Testing extreme value models. Extremes 3, 363–384 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    MathSciNet  Google Scholar 

  • Öztürk, A., Korukoglu, S.: A new test for the extreme value distribution. Comm. Statist. Simulation Comput. 17, 1375–1393 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  • Peng, L.: Estimation of the coefficient of tail dependence in bivariate extremes. Stat. Probab. Lett. 43, 399–409 (1999)

    Article  MATH  Google Scholar 

  • Peng, L.: Bias-corrected estimators for monotone and concave frontier functions. J. Statist. Plann. Inf. 119, 263–275 (2004)

    Article  MATH  Google Scholar 

  • Ramesh, N.I., Davison, A.C.: Local models for exploratory analysis of hydrological extremes. J. Hydrology 256, 106–119 (2002)

    Article  Google Scholar 

  • Ramos, A., Ledford, A.: Regular score tests of independence in multivariate extreme values. Extremes 8, 5–26 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • Shi, D.J.: An analysis of variance test for the extreme value distribution. J. Tianjin Univ. 2, 116–121 (1988)

    Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

  • 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)

    Google Scholar 

  • 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)

    Article  MATH  MathSciNet  Google Scholar 

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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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