% IMPORTANT: The following is UTF-8 encoded. This means that in the presence
% of non-ASCII characters, it will not work with BibTeX 0.99 or older.
% Instead, you should use an up-to-date BibTeX implementation like “bibtex8” or
% “biber”.
@PHDTHESIS{Johnen:795979,
author = {Johnen, Marcus},
othercontributors = {Kamps, Udo and Kateri, Maria},
title = {{C}ontributions to statistical inference based on
sequential order statistics from exponential and {W}eibull
distributions},
school = {RWTH Aachen University},
type = {Dissertation},
address = {Aachen},
reportid = {RWTH-2020-08694},
pages = {1 Online-Ressource (vi, 165 Seiten) : Illustrationen,
Diagramme},
year = {2020},
note = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
University; Dissertation, RWTH Aachen University, 2020},
abstract = {In reliability theory and applications, modeling the
lifetimes of technical systems with several components plays
an important role. For instance, interest may lie in
describing the lifetimes of components within k-out-of-n
systems which consist of n identical components and work as
long as at least k of these components are running. In many
such systems, the remaining components experience an
increased load after some component has failed. This effect,
also called load-sharing effect, can be modeled, e.g., by
sequential order statistics which were introduced as a
generalization of common order statistics. In this thesis, a
sub-model ensuring proportional hazard rates is examined for
the case of an underlying exponential or Weibull
distribution. Here, the focus lies on questions regarding
the fitting of this model to given data. More detailed,
topics of classical statistical inference such as point
estimation, hypothesis testing, and confidence sets are
discussed. To this end, the underlying structure of
transformation models proves helpful which gain much
attention in the literature next to the theory of
exponential families. For the model of sequential order
statistics with an underlying exponential distribution, the
transformation model approach leads to the minimum risk
equivariant estimators of the model parameters as an
alternative to the known maximum likelihood estimator (MLE)
or the uniformly minimum variance unbiased estimator. In
addition, we present a method to derive exact
goodness-of-fit tests on this model. For the model with an
underlying Weibull distribution, we start by proving certain
regularity conditions which lead to the Fisher information
matrix and which, eventually, implicate the consistency and
asymptotic efficiency of the MLE of the model parameters.
Moreover, the MLE is seen to satisfy certain pivotal
properties where the distributions of several quantities
comprising the estimator and the parameters are independent
of the true underlying parameters. These properties are, in
fact, shown to be a consequence of the equivariance of the
MLE, which also proves them for a much larger class of
estimators. We demonstrate that the MLE of the shape
parameter of the underlying Weibull distribution is biased
and subsequently discuss several methods for reducing this
bias, leading to a variety of other equivariant estimators.
By means of simulation and by utilizing the pivotal
properties mentioned earlier, these alternative estimators
are shown to be superior in terms of variance and mean
squared error as well. Different null hypotheses, e.g. for
testing the adequacy of one particular model or the presence
of a load-sharing effect, are discussed. Here, three
well-known test statistics given by the likelihood ratio,
Rao’s score, and Wald’s statistic are applied which
usually lead to asymptotic tests. However, the
transformation model structure allows for the derivation of
exact tests based on these statistics which can also be
compared much easier via simulations. Thereafter, exact and
asymptotic confidence sets for the Weibull shape parameter
and for the model parameters of the sequential order
statistics are addressed. In the case where the model
parameters are known and the Weibull shape parameter is the
only unknown parameter, the resulting univariate
log-likelihood function may have multiple local maxima which
might lead to problems when trying to find the MLE. This
circumstance is further analyzed and a possible solution is
addressed. We observe that, other than in a similar and
well-known situation concerning samples from a Cauchy
distribution with an unknown location parameter, this
problem is seen to be caused by the corresponding
Kullback-Leibler divergence having multiple local minima.
Finally, two approaches are proposed to generalize the model
of sequential order statistics from an underlying Weibull
distribution to other distributions that stem from a
log-location-scale family of distributions. Here, several
properties are seen to be maintained during this transition
depending on whether the transformation model structure or a
proportional hazard rate property are conserved. For
illustration, the methods derived in this thesis are applied
to two real data sets discussed in the literature.},
cin = {116410 / 110000},
ddc = {510},
cid = {$I:(DE-82)116410_20140620$ / $I:(DE-82)110000_20140620$},
typ = {PUB:(DE-HGF)11},
doi = {10.18154/RWTH-2020-08694},
url = {https://publications.rwth-aachen.de/record/795979},
}
guest ::