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

where Ẽj = Z j − t if Z j > t and 1 − F GP D (x) = ( )
1 + γ −
1
1x γ 1
. Then, the maximization of
σ
this expression leads to a POT estimator for γ 1 which we further denote by ˆγ t,ML.
(c)
Secondly, we can construct a new estimator based on k upper order statistics for instance
within the framework of the QQ-plot regression technique. For example, in the case of
no-censoring, Beirlant et al. (1996) proposed an estimator of a real-valued index based on
a generalized quantile plot, which takes over the role of the Pareto quantile plot in this
more general setting. More precisely they proposed to look at the graph with coordinates
( n + 1
)
log , log UH j,n , j = 1, ..., n − 1,
j
with UH j,n = X n−j,n H X,j,n . Again this plot becomes ultimately linear for small j with
slope approximating γ 1 . Then, one can construct several regression based estimators, such
as
ˆγ k,UH = 1 k∑
log UH j,n − log UH k+1,n .
k
j=1
From the above it appears natural to define a generalization of ˆγ k,UH to the censoring
case as a slope estimator of the generalized quantile plot adapted for censoring
( (
− log 1 − ˆFn (Z n−j+1,n ) ) , log UH j,n) (c) , (10)
(j = 1, ..., n − 1) where UH (c)
j,n = Z n−j,n H (c)
Z,j,n:
ˆγ (c)
1 ∑ kj=1
k,UH = log UH (c)
k
1
k
j,n − log UH (c)
k+1,n
∑ kj=1
. (11)
δ n−j+1,n
Using one of the abovementioned estimators ˆγ (c)
.,. of γ 1 ≥ 0 we can now propose new
estimators for the quantile x F,p , in the spirit of the one proposed by Dekkers et al. (1989)
in the case of no-censoring:
ˆx (c)
p,t,. = t + ˆγ (c)
.,. t
( 1− ˆFn(t)
)ˆγ (c)
.,.
p
ˆγ (c)
.,.
− 1
. (12)
Under suitable assumptions, we establish the asymptotic properties of our estimators. We
illustrate their behaviour in a small simulation study, but also in a practical insurance
example.
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