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Abstract:
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First characterizations of probability distributions date to the
thirties of last century. This area, which lies on the borderline of probability
theory and mathematical statistics, attracts large number of researchers, and
in recent times the number of papers on the subject is increasing.
Goodness-of- t tests are among the most important nonparametric tests.
Many of them are based on empirical distribution function. The application
of characterization theorems for construction of goodness-of- t tests dates
to the middle of last century, and recently has become one of the main
directions in this eld. The advantage of such tests is that they are often free
of distribution parametres and hence enable testing of composite hypotheses.
The goals of this dissertation are the formulation of new characterizations
of exponential and Pareto distribution, as well as the application of the theory
of U-statistics, large deviations and Bahadur e ciency to construction
and examination of asymptotics of goodness-of- t tests for aforementioned
distributions. The dissertation consists of six chapters.
In the rst chapter a review of di erent types of characterizations is presented,
pointing out their abundance and variety. The special emphasis is
given to the characterizations based on equidistribution of functions of the
sample. Besides, two new characterizations of Pareto distribution are presented.
The second chapter is devoted to some new characterizations of the exponential
distributions presented in papers [65] and [53]. Six characterizations
based on order statistics are presented. A special case of one of them (theorem
2.4.3) represents the solution of open problem stated by Arnold and
Villasenor [9].
In the third chapter there are basic concepts on U-statistics, the class
of statistics important in the theory of unbiased estimation. Some of their
asymptotic properties are given. U-empirical distribution functions, a generalization
of standard empirical distribution functions, are also de ned. The
fourth chapter is dedicated to the asymptotic e ciency of statistical tests,
primarily to Bahadur asymptotic e ciency, i.e. asymptotic e ciency of the
test when the level of signi cance approaches zero. Some theoretical results
from the monograph by Nikitin [57], and papers [61], [59], etc. are shown.
In the fth chapter new results in the eld of goodness-of- t tests for
Pareto distribution are presented. Based on three characterizations of Pareto
distribution given in section 1.1.2. six goodness-of- t tests, three of integral,
and three of Kolmogorov type, are proposed. In each case the composite
null hypothesis is tested since the test statistics are free of the parameter
of Pareto distribution. For each test the asymptotic distribution under null
hypothesis, as well as asymptotic behaviour of the tail (large deviations) under
close alternatives is derived. For some standard alternatives, the local
Bahadur asymptotic e ciency is calculated and the domains of local asymptotic
optimality are obtained. The results from this chapter are published in
[66] and [64].
The sixth chapter brings new goodness-of- t tests for exponential distribution.
Based on the solved hypothesis of Arnold and Villasenor two
classes of tests, integral and Kolmogorov type, are proposed, depending on
the number of summands in the characterization. The study of asymptotic
properties, analogous to the ones in the fth chapter is done in case of two
and three summands, for which the tests have practical importance. The
results of this chapter are presented in [39]. |