Estimation of the extreme value index and high quantiles under ...

leading to the estimator
H (c)
Z,t =
∑ ni=1
log(Z i /t)1l {Zi >t}
∑ ni=1
, (6)
δ i 1l {Zi >t}
while for the extreme quantile estimator we propose to use
ˆx (c)
p,t = t
( 1 − ˆFn (t)
p
) H
(c)
Z,t
, (7)
where ˆF n (x), −∞ < x < τ H denotes the Kaplan-Meier (1958) product limit estimator of
F (x), defined as
1 − ˆF
n∏
[
n (x) = 1 − δ ]
j,n1l Zj,n ≤x
,
n − j + 1
j=1
where Z j,n denote the order statistics associated to Z 1 , ..., Z n and δ j,n := δ k if and only if
Z j,n = Z k .
The corresponding tail probability estimator is now of course given by
IP ˆ
(c) (X > x) = (1 − ˆF n (t)) ( x) (c) −1/H Z,t
. (8)
t
When choosing t = Z n−k,n , we obtain the estimator
H (c)
Z,k,n = ∑ kj=1
(
log(Zn−j+1,n ) − log(Z n−k,n ) )
∑ kj=1
δ n−j+1,n
, (9)
which is the original Hill estimator adapted for right censoring.
We will give also another interpretation for this estimator which is based on a novel
QQ-plot.
2.2. Observing (Z, δ), X independent of Y , X in the domain of attraction of
the Fréchet or Gumbel law, and Y in the domain of attraction of the Fréchet
law
When considering the extension to the case where γ 1 ≥ 0, again as in the no-censoring case
there are mainly two sets of solutions which originated from two different formulations of
the model.
First, the maximum likelihood approach based on POT’s (Peaks over Threshold) is based
on the results given by Balkema and de Haan (1974) and Pickands (1975), stating that
the limit distribution of the absolute exceedances over a threshold t when t → ∞ is given
by a generalized Pareto distribution (GPD). In the case of censoring, we can easily adapt
the likelihood to
k∏ [
fGP D (Ẽj) ] δ j
[
1 − FGP D (Ẽj) ] 1−δ j
j=1
4