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@PHDTHESIS{Su:1024857,
author = {Su, Zhe},
othercontributors = {Kamps, Udo and Kateri, Maria},
title = {{M}ixture distributions as exponential families},
school = {RWTH Aachen University},
type = {Dissertation},
address = {Aachen},
publisher = {RWTH Aachen University},
reportid = {RWTH-2026-00377},
pages = {1 Online-Ressource : Illustrationen},
year = {2026},
note = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
University; Dissertation, RWTH Aachen University, 2026},
abstract = {In many real-world applications, datasets contain a mixture
of different types of information. For example, a dataset
may combine continuous and discrete components or contain a
large proportion of zeros alongside a distribution of
positive values. Such mixed structures are common in fields
such as engineering, ecology, biostatistics, finance, and
reliability studies. In such situations, classical
single-component distributions may not describe the data
well, and mixture distributions provide a useful
alternative. One phenomenon that has received considerable
attention is zero-inflation, where the frequency of zeros in
the data exceeds what is expected under a chosen model
distribution. In case of a continuous distribution, the
probability of zeros is zero, but zeros may be observed in a
real-world application. To capture this feature,
zero-inflated models were developed. These are typically
formulated as two-component mixtures: one component is a
degenerate distribution at zero, and the other is a
non-degenerate probability distribution. If the latter
belongs to an exponential family, then, with suitable
parameterizations, the resulting zero-inflated model can
form an exponential family, too. This aspect has received
only limited attention in the existing literature.
Exponential families and, more restrictively, regular
exponential families exhibit particular favorable properties
in statistical analysis. When a regular exponential family
is present, a canonical representation leads directly to
minimal sufficient and complete statistics and smooth mean
value and cumulant functions, which facilitate the
derivation of uniformly minimum-variance unbiased
estimators, the characterization of the existence and
uniqueness of maximum likelihood estimators, and the study
of their asymptotic behavior. Moreover, a regular
exponential family allows for the construction of optimal
tests and optimal confidence intervals in a natural and
elegant way. This doctoral thesis studies a broad class of
two-component mixture models which form regular exponential
families. The starting point is a formulation with
functionally unrelated parameters; under appropriate
assumptions, this formulation is shown to yield a regular
exponential family. By imposing functional relationships
among the parameters within a single component or across
components, or by fixing some parameters, regular
exponential families of lower orders are derived. The
developed construction approaches apply not only to
zero-inflated and zero-altered models, but also to a much
wider range of two-component mixtures, including models with
multiple inflation or alteration points. Statistical
inference within the two-component mixtures as regular
exponential families is then studied. Maximum likelihood
estimation is analyzed, with particular emphasis on
conditions ensuring existence and uniqueness. In parallel,
uniformly minimum-variance unbiased estimators are derived,
including explicit expressions for the mixture parameter and
the distribution mean under appropriate assumptions. Several
testing problems are examined in detail. Among them is the
assessment of whether a two-component mixture is required in
place of a single-component model. Four scenarios are
considered in this context. The first concerns models in
which certain mixture distributions can be viewed as the
untruncated form of one of their components, with the
testing problem being whether the simpler untruncated model
should be used instead. The second handles discrete models
with multiple-point alterations, where a likelihood ratio
test is developed; within the same framework, a uniformly
most powerful unbiased test is established for testing
alteration at a single point. The third scenario focuses on
discrete models with one-point alteration, for which
uniformly most powerful unbiased tests are derived to assess
alteration, inflation, or deflation at that point. The
fourth scenario considers mixtures of a discrete and a
continuous distribution, and a uniformly most powerful test
is constructed for this case. An important advantage of
regular exponential families is that the existence of most
powerful tests can often be stated directly. Both uniformly
most powerful and uniformly most powerful unbiased tests for
selected model parameters are derived, taking into account
scenarios with and without functional restrictions on
parameters. Utilizing the duality between confidence
intervals and hypothesis tests, corresponding most accurate
confidence intervals are obtained, and the characterization
of interval bounds is discussed. The theoretical framework
is illustrated through three applications: a zero-altered
Poisson model, a zero-to-m-altered Poisson model, and a
geometric-exponential mixture model. Extensive simulation
studies demonstrate the practical performance of the
proposed inference methods. Finally, the scope of the
analysis is broadened to include a class of two-component
mixtures that are exponential families but not necessarily
regular. Particular attention is given to the full-rank but
non-regular scenario, in which favorable inferential
properties, similar to those in the regular case, are
present and demonstrated using three concrete examples.},
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-2026-00377},
url = {https://publications.rwth-aachen.de/record/1024857},
}
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