# 52413

Автор(ов):

2

Параметры публикации
Тип публикации:
Тезисы доклада
Название:
Discrepancy method for extremal index estimation
Электронная публикация:
Да
Наименование конференции:
11th international conference on Extreme Value Analysis (EVA 2019, Zagreb)
Наименование источника:
Book of abstracts of the 11th international conference on Extreme Value Analysis (EVA 2019, Zagreb)
Город:
Загреб
Издательство:
Университет Загреб
Год издания:
2019
Страницы:
79-80
Аннотация
Let fXngn1 be a sequence of random variables (r.v.s) with a cumulative distribution function (cdf) F(x); denote Mn = maxfX1; :::;Xng: Recall, that the stationary sequence fXngn1 is said to have the extremal index (EI)  2 [0; 1] if for each 0 <  < 1 there is a sequence of real numbers un = un( ) such that limn!1 n(1 􀀀 F(un)) =  and limn!1 PfMn  ung = e􀀀 hold. The problem is that nonparametric estimators of EI require usually the choice of a threshold parameter u and/or a declustering parameter b or r, . Our aim is to propose a nonparametric tool to nd one parameter, e.g., u, b or r, by samples of moderate sizes. For this purpose, we extend the so-called discrepancy method, , for EI estimation in case of heavy-tailed distributions. The idea of this method is to nd an unknown parameter h of the cdf as a solution of the discrepancy equation (Fh; b Fn) = ; where b Fn is an empirical cdf of a sample and (; ) is a metric in the space of cdf's. Following , dene T1(u) = minfj  1 : M1;j  u;Xj+1 > ujX1 > ug; where M1;j = maxfX2; :::;Xjg, M1;1 = 􀀀1. Observations of T1(un) normalized by the tail function fYi = F(un)T1(un)ig, i = 1; :::;L, L = L(un), are such that PfF(un)T1(un) > tg !  exp(􀀀t) =: 1 􀀀 G(t; ); t > 0; as n ! 1 under a specic mixing condition, . In our work we propose a normalization of the von Mises-Smirnov (M-S) statistic !2n and use it as a metric  in the discrepancy equation to select the optimal value of a parameter. Denote for some estimator ^(u) of EI !2 k(^(u)) = Xk􀀀1 i=0 G(Y(L􀀀i); ^(u)) 􀀀 G(Y(L􀀀k); ^(u)) 1 􀀀 G(Y(L􀀀k); ^(u)) 􀀀 k 􀀀 i 􀀀 0:5 k !2 + 1 12k : (2) A value of u can be found as a solution of the discrepancy equation with regard to any consistent nonparametric estimator of EI. The calculation of (2) by the entire sample may lead to the lack of a solution of the discrepancy equation regarding u: The selection of k and  remains a problem. To overcome this problem we prove that the limit distribution of !2 k(^(u)) is the same as for the M-S statistic. Then its quantiles may be used as . Theorem 1 Let L 􀀀 k ! 1; k ! 1 and the estimator of EI b = bmn be such that p mn(bmn 􀀀 ) !d  as n ! 1; where the r.v.  has a non-degenerate cdf F. Assume that the sequence mn is such that k=mn = o(1) and (lnL)2=mn = o(1) as n ! 1: Then ^!2 k(bmn) !d   A1 holds, where A1 is the limit distribution function of the M-S statistic. References 79  J. Beirlant, Y. Goegebeur, J. Teugels, J. Segers, Statistics of Extremes: Theory and Applications, Wiley, 2004  V. N. Vapnik, N.M. Markovich, A.R. Stefanyuk, Rate of convergence in L2 of the projection estimator of the distribution density. Automat. Rem. Contr., 1992  C.A.T. Ferro, J. Segers, Inference for Clusters of Extreme Values. J. R. Statist. Soc. B, 2003
Библиографическая ссылка:
Маркович Н.М., Родионов И.В. Discrepancy method for extremal index estimation / Book of abstracts of the 11th international conference on Extreme Value Analysis (EVA 2019, Zagreb). Загреб: Университет Загреб, 2019. С. 79-80.