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], 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, [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)) =
Xk1
i=0
G(Y(Li); ^(u)) G(Y(Lk); ^(u))
1 G(Y(Lk); ^(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
