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 fXngn1 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 fXngn1 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, [1]. 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, [2], 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 [3], dene 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 specic mixing condition, [3]. 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 = bmn be such that p mn(bmn 􀀀 ) !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(bmn) !d   A1 holds, where A1 is the limit distribution function of the M-S statistic. References 79 [1] J. Beirlant, Y. Goegebeur, J. Teugels, J. Segers, Statistics of Extremes: Theory and Applications, Wiley, 2004 [2] 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 [3] 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.