Abstract
Consider an iid sampleZ 1,...,Z n with common distribution functionF on the real line, whose upper tail belongs to a parametric family {F β: β∈⊝}. We establish local asymptotic normality (LAN) of the loglikelihood process pertaining to the vector(Z n−i+1∶n ) k i=1 of the upperk=k(n)→ n→∞∞ order statistics in the sample, if the family {F β:β∈⊝} is in a neighborhood of the family of generalized Pareto distributions. It turns out that, except in one particular location case, thekth-largest order statisticZ n−k+1∶n is the central sequence generating LAN. This implies thatZ n−k+1∶n is asymptotically sufficient and that asymptotically optimal tests for the underlying parameter β can be based on the single order statisticZ n−k+1∶n . The rate at whichZ n−k+1∶n becomes asymptotically sufficient is however quite poor.
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References
Balkema, A. A. and de Haan, L. (1974). Residual life time at great age,Ann. Probab.,2, 792–804.
Castillo, E., Galambos, J. and Sarabia, J. M. (1989). The selection of the domain of attraction of an extreme value distribution from a set of data,Extreme Value Theory (eds. J. Hüsler and R.-D. Reiss),Lecture Notes in Statist,51, 181–190, Springer, New York.
Csörgő, S. and Mason, D. M. (1985). Central limit theorems for sums of extreme values,Mathematical Proceedings of the Cambridge Philosophical Society,98, 547–558.
Dekkers, A. L. M. and de Haan, L. (1989). On the estimation of the extreme-value index and large quantile estimation,Ann. Statist. 17, 1795–1832.
Falk, M. (1992). On testing the extreme value index via the POT-method,Ann. Statist. (to appear).
Falk, M., Hüsler, J. and Reiss, R.-D. (1994).Laws of Small Numbers: Extremes and Rare Events, DMV Seminar 23, Birkhäuser, Basel.
Galambos, J. (1987).The Asymptotic Theory of Extreme Order Statistics (2nd ed.), Krieger, Malabar.
Gnedenko, B. (1943). Sur la distribution limité du terme maximum d'une série aléatoire,Ann. Math.,44, 423–453.
Gumbel, E. J. (1958).Statistics of Extremes, Columbia University Press, New York.
Hall, P. and Welsh, A. H. (1985) Adaptive estimates of parameters of regular variation,Ann. Statist.,13, 331–341.
Hasofer, A. M. and Wang, Z. (1992). A test for extreme value domain of attraction,J. Amer. Statist. Assoc.,87, 171–177.
Hill, B. M. (1975). A simple approach to inference about the tail of a distribution,Ann. Statist.,3, 1163–1174.
Hosking, J. R. M. and Wallis, J. R. (1987). Parameter and quantile estimation for the generalized Pareto distribution.Technometrics,29, 339–349.
Janssen, A. and Marohn, F. (1994). On statistical information of extreme order statistics, local extreme value alternatives, and Poisson point processes,J. Multivariate Anal.,48, 1–30.
Janssen, A. and Reiss, R.-D. (1988). Comparison of location models of Weibull type samples and extreme value processes,Probab. Theory Related Fields,78, 273–292.
LeCam, L. (1960). Locally asymptotically normal families of distributions,University of California Publications in Statistics.3, 37–98.
LeCam, L. (1986). Asymptotic methods in statistical decision theory.Springer Ser. Statist., Springer, New York.
LeCam, L. and Yang, G. L. (1990). Asymptotics in statistics (some basic concepts),Springer Ser. Statist. Springer, New York.
Marohn, F. (1991). Global sufficiency of extreme order statistics in location models of Weibull type,Probab. Theory Related Fields,88, 261–268.
Marohn, F. (1994a). On testing the exponential and Gumbel distribution,Extreme Value Theory and Applications (eds. J. Galambos, J. Lechner and E. Simiu), 159–174. Kluwer Dordrecht.
Marohn, F. (1994b). Asymptotic sufficiency of order statistics for almost regular Weibull type densities,Statist. Decisions,12, 385–393.
Marohn, F. (1995). Neglecting observations in Gaussian sequences of statistical experiments,Statist. Decisions,13, 83–92.
Pickands, J., III (1975). Statistical inference using extreme value order statistics,Ann. Statist.,3, 119–131.
Reiss, R.-D. (1989). Approximate distributions of order statistics (with applications to nonparametric statistics).Springer Ser. Statist., Springer, New York.
Reiss, R.-D., (1993). A course on point processes,Springer Ser. Statist., Springer, New York.
Smith, R. L. (1987). Estimating tails of probability distributions,Ann. Statist.,15, 1174–1207.
Strasser, H. (1985). Mathematical theory of statistics,De Gruyter Stud. Math.,7, De Gruyter, Berlin.
Wei, X. (1992). Asymptotically efficient estimation of the index of regular variation, Ph.D. thesis, Department of Statistics, University of Michigan.
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Falk, M. LAN of extreme order statistics. Ann Inst Stat Math 47, 693–717 (1995). https://doi.org/10.1007/BF01856542
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DOI: https://doi.org/10.1007/BF01856542