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Abstract:
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The main objective of this dissertation is to develop novel goodness-of-
fit tests based on independence characterizations and to investigate their proper-
ties. The starting point is the fact that certain distributions can be uniquely char-
acterized by the independence of suitable functions of random variables. This
observation makes it possible to reduce goodness-of-fit testing for a hypothe-
sized model to the problem of detecting departures from the corresponding in-
dependence condition. Accordingly, the proposed test statistics are constructed
by comparing empirical versions of joint functionals with the product of the cor-
responding marginal functionals, where the resulting discrepancy serves as the
basis for new goodness-of-fit tests for different classes of distributions.
One of the contributions of the dissertation is the development of goodness-
of-fit tests for the geometric distribution based on Ferguson’s independence char-
acterization. In this setting, the general framework of comparing joint and
marginal empirical functionals is implemented through V-empirical probability
generating functions. Furthermore, a novel approach to goodness-of-fit testing
for absolutely continuous distributions is proposed, relying on the discrepancy
between the joint U-empirical distribution function of an appropriately defined
random vector and the product of its marginal U-empirical distribution func-
tions. Another part of the research focuses on circular data. A goodness-of-fit test
for the wrapped normal distribution is proposed, where the general methodology
is adapted to the periodic structure of observations on the circle.
The theoretical part of the dissertation is devoted to the derivation of the
asymptotic properties of the proposed test statistics, drawing on the theory of
U- and V- statistics as well as on the theory of U-empirical processes. An addi-
tional contribution is the establishment of new asymptotic results for finite linear
combinations of weakly degenerate V-statistics, both in settings with known pa-
rameters and in settings involving parameter estimation. These findings provide
the theoretical foundation for a class of combined tests in which appropriately
chosen weights regulate the contribution of individual components and can be
used to enhance the overall efficiency of the testing procedure.
The practical performance of the proposed methods will be assessed through
extensive simulation studies and the analysis of several real data sets. The sim-
ulation study will investigate the finite-sample properties of the tests under the
null hypothesis and a range of relevant alternatives, while the real-data applica-
tions will demonstrate their usefulness in practical statistical problems. In this
way, the empirical part of the dissertation will complement the theoretical de-
velopments and provide evidence of the practical applicability of the proposed
methodologies. |