Abstract
We consider an i.i.d. sample, generated by some distribution function, which belongs to the domain of attraction of an extreme value distribution with unknown shape and scale parameters. We treat the scale parameter as a nuisance parameter and establish for the hypothesis of Gumbel domain of attraction an asymptotically optimal test based on those observations among the sample, which exceed a given threshold sequence. Asymptotic optimality is achieved along certain contiguous extreme value alternatives within the concept of local asymptotic normality (LAN). Adaptive test procedures exist under restrictive assumptions. The finite sample size behavior of the proposed test is studied by simulations and it is compared to that of a test based on the sample coefficient of variation.
Similar content being viewed by others
References
Balkema, A.A. and de Haan, L., “Residual life time at great age,” Ann. Probab. 2, 792–804, (1974).
Bardsley, W.E., “A test for distinguishing between extreme value distributions,” J. Hydrology 34, 377–381, (1977).
Bartholomew, D.J., “Testing for departure from the exponential distribution,” Biometrica 44, 253–257, (1957).
Castillo, E., Galambos, J., and Sarabia, J.M., “The selection of the domain of attraction of an extreme value distribution from a set of data,” In: Extreme Value Theory (Proc. Oberwolfach, 1987), J. Hüsler and R.-D. Reiss, eds. Lecture Notes in Statistics 51, Springer, New York, 181–190, (1989).
Falk, M., “On testing the extreme value index via the POT-method,” Ann. Statist. 23, 2013–2035, (1995a).
Falk, M., “LAN of extreme order statistics,” Ann. Inst. Statist. Math. 47, 693–717, (1995b).
Falk, M., Hüsler, J., and Reiss, R.-D., Laws of Small Numbers: Extremes and Rare Events, DMV Seminar Vol. 23, Birkhäuser, Basel, 1994.
Galambos, J., “A statistical test for extreme value distributions,” In: Non-Parametric Statistical Inference, B.V. Gnedenko et al., eds, North Holland, Amsterdam, 221–230, (1982).
Gomes, M.I., “Comparison of extremal models through statistical choice in multidimensional backgrounds,” In: Extreme Value Theory (Proc. Oberwolfach, 1987) J. Hüsler and R.-D. Reiss, eds.), Lecture Notes in Statistics 51, Springer, New York, 191–203, (1989).
Gomes, M.I. and Alpuim, M.T., “Inference in a multivariate generalized extreme value model-asymptotic properties of two test statistics,” Scand. J. Statist. 13, 291–300, (1986).
Gomes, M.I. and Van Montfort, M.A.J., “Exponentiality versus generalized Pareto-quick tests,” Proc. III Internat. Conf. Statistical Climatology, 185–195, (1986).
Hasofer, A.M. and Wang, Z., “A test for extreme value domain of attraction,” J. Americ. Statist. Assoc. 87, 171–177, (1992).
Hosking, J.R.M., “Testing whether the shape parameter is zero in the generalized extreme value distribution,” Biometrika 71, 367–374, (1984).
Janssen, A., Milbrodt, H., and Strasser, H., Infinitely Divisible Statistical Experiments, Lecture Notes in Statisics 27. Springer, Berlin-Heidelberg, 1985.
LeCam L., “Locally asymptotically normal families of distributions,” University of California Publications in Statistics 3, 37–98, (1960).
LeCam, L., Asymptotic Methods in Statistical Decision Theory, Springer Series in Statistics, New York, 1986.
LeCam, L. and Yang, G.L., Asymptotics in Statistics. Some Basic Concepts, Springer Series in Statistics, New York, 1990.
Marohn, F., “On testing the exponential and Gumbel distribution,” In: Extreme Value Theory (J. Galambos et al., eds), Kluwer, Dordrecht, 159–174, (1994).
Marohn, F., “Contributions to a local approach in extreme value statistics,” Habilitation thesis, Katholische Universitaät Eichstaätt, 1995.
Marohn, F., “An adaptive efficient test for Gumbel domain of attraction,” Scand. J. Statist. 25, 311–324 (1998).
Otten, A. and Van Montfort, M.A.J., “The power of two tests on the distribution of extremes,” J. Hydrology 37, 195–199, (1978).
Reiss, R.-D., A Course on Point Processes, Springer Series in Statistics, New York, 1993.
Resnick, S., Extreme Values, Regular Variation, and Point Processes, Springer, New York, 1987.
Rychlik, T., “Weak limit theorems for stochastically largest order statistics,” In: Order statistics and Nonparametrics: Theory and Applications, (P.K. Sen and I.A. Salama, eds), North Holland, Amsterdam, 141–154, (1992).
Strasser, H., Mathematical Theory of Statistics, De Gruyter, Berlin-New York, 1985.
Van Montfort, M.A.J., “On testing that the distribution of extremes is of type I when type II is the alternative,” J. Hydrology 11, 421–427, (1970).
Van Montfort, M.A.J. and Gomes, M.I., “Statistical choice of extremal models for complete and censored data,” J. Hydrology 77, 77–87, (1985).
Van Montfort, M.A.J. and Witter, J.V., “Testing exponentiality against generalized Pareto distribution,” J. Hydrology 78, 305–315, (1985).
Weissman, I., “Estimation of parameters and large quantiles based on the k largest observations”, J. Americ. Statist. Assoc. 73, 812–815, (1978).
Witting, H., Mathematische Statistik I, Teubner, Stuttgart, 1985.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Marohn, F. Testing the Gumbel Hypothesis Via the Pot-Method. Extremes 1, 191–213 (1998). https://doi.org/10.1023/A:1009910806693
Issue Date:
DOI: https://doi.org/10.1023/A:1009910806693