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Abstract:
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Goodness of t and symmetry tests occupy a signi cant part
of nonparametric statistic. Most of classical tests are based on the distance
between the assumed distribution function and its consistent estimate, empirical
distribution function. The symmetry tests are analogously constructed.
A new approach that is especially attractive in recent years is making tests
based on characterizations of di erent types. Those tests use U-empirical
distribution functions (generalized empirical ones), U-empirical transforms
(eg. Laplace transform, characteristic functions etc.) and U-empirical moments
of distributions. The main advantage of these tests is that they are
often free of some distribution parameters. Therefore they are suitable for
testing composite hypothesis.
For purpose of comparison of tests the Bahadur e ciency has become
very popular. One of the reasons is that it does not require the asymptotic
normality of test statistics. In addition, Bahadur and Pitman e ciencies very
often locally coincide. It turns out that for determining Bahadur e ciency it
is necessary to nd large deviations function under null hypothesis. If that is
not possible, usually the approximate Bahadur e ciency is used. It requires
the existence of asymptotic distribution and the asymptotic behavior of its
tail under null hypothesis, and the limit in probability under alternative
distribution.
The goals of the thesis are the construction of new goodness of t and
symmetry tests based on U-statistics and V -statistics, deriving the asymptotic
distribution of proposed statistics, their large deviation functions and
Bahadur e ciencies or their approximations. The thesis is divided into two
parts.
The rst part consists of three chapters. In the rst chapter the theory
of U-statistics and V -statistics as well as U-empirical and V -empirical distribution
function and some other empirical transforms is presented. The
second chapter is devoted to the U-statistics and V -statistics with estimated
parameters. The third chapter deals with asymptotic e ciency of nonparametric
tests. Most of the chapter is devoted to Bahadur e ciency. In the
same chapter the large deviation function for a new class of tests statistics
is derived. This result is presented in [69].
The second part, which starts with the fourth chapter, is dedicated to new
tests based on U-statistics and V -statistics. In the fourth chapter some type
of characterizations which are used within the next chapters for construction
of tests are presented. They include characterizations based on equidistribution
of some statistics among which the characterizations of symmetric
distributions stand out, then those based on functional equations that the
distribution function satis es, those based on the independence of statistics
and those based on moments. Two new characterizations of symmetric distributions
are also presented. The fth chapter deals with new goodness of t
tests. There are four new exponentiality tests, two new goodness of t tests
for a family of Pareto distribution, as well as two new goodness of t tests
for logistic distribution (see [66], [69]). Also one new class of uniformity tests
which can be used as goodness of t test for any predetermined continuous
distribution is proposed (see [67]). The sixth chapter is devoted to new symmetry
tests. Five new symmetry tests are proposed. In the seventh chapter
there are new exponentiality tests based on U-statistics and V -statistics with
estimated parameters. A special attention is given to new tests based on U-
empirical Laplace transforms. Beside those, some tests based on U-empirical
moments are also presented. For new presented tests local Bahadur and/or
Pitman e ciency is calculated. In the nal chapter, a brief review of some
applications in time series analysis is shown. |