|
Abstract:
|
Statistical methodology for dealing with extremes depend on how
extreme values are defined. One way to extract extremes from a given sample
x1, x2, ..., xN is to consider maxima (minima). The other way is to consider
values y1 = x1 − u, y2 = x2 − u, . . . , yn = xn − u, where y1, y2, . . . , yn are
sample members above (below) a given predetermined threshold u. These
two methods lead to two different approaches in extreme value theory.
This doctoral dissertation has two main goals. One of them is to apply the
techniques from extreme value framework to certain type of combinatorial
problems. The other goal is to contribute to the field of statistical modeling
of extremes. The dissertation consists of three chapters.
In the first chapter, we introduce generalized extreme value distributions
and generalized Pareto distributions (GPD). These two families play key
roles in the two approaches to modeling extremes. We set out the theoretical
background for both approaches.
In the second chapter, we apply the extremal techniques to combinatorial
waiting time problems. Precisely, we consider Coupon collector’s problem,
defined as follows: elements are sampled with replacement from the
set Nn = {1, 2, . . . , n} under assumption that each element has probability
1/n of being drawn. The subject of interest is the waiting time Mn until all
elements of Nn or some other pattern are sampled. We focus our attention
to the following two cases:
1. Mn is the waiting time until all elements of Nn are sampled at least r
times, where r is a positive integer;
2. Mn is the waiting time until all pairs of elements jj, j ∈ Nn are sampled.
We present new results related to the asymptotic behavior of the waiting
time Mn, if it is known that a large number of trials was performed and
the experiment is not over. For both cases, we determine the limiting distribution
of exceedances of Mn over high thresholds, and answer some related
questions: how to choose a suitable high threshold (depending on n) in order
to obtain a limiting distribution; under what conditions the limit does not
depend on the threshold; are the generalized Pareto distributions the only
possible limits. We also estimate the speed of convergence in both cases.
The third chapter of the dissertation is devoted to estimation of parameters
and quantiles of the generalized Pareto distributions. We restrict the
attention to the two-parameter version of GPD, defined as:
Wγ,σ(x) =


1 − e−x
, x ≥ 0, γ = 0
1 −
1 + γ
σx
−1
, x ≥ 0, γ > 0
1 −
1 + γ
σx
−1
, x ∈
h
0,−σ
γ
i
, γ < 0.
Well known problem with this model is inconsistency with the sample
data, which is that one or more sample observations exceed the estimated
upper bound in case when γ < 0. We propose a new, general technique
to overcome the inconsistency problem and improve performance of the existing
GPD estimation methods. We apply the proposed technique to methodof-
moments and method-of-probability-weighted-moments estimates, investigate
its performance through computer simulation and provide some real data
examples. Finally, we address the problem of estimating high GPD quantiles.
We evaluate the robustness of some estimation methods through simulation
study and present a case study from finance (value-at-risk estimation), with
special emphasis to certain difficulties related to this field of application. |