# ASIMPTOTSKA SVOJSTVA NEPARAMETARSKIH TESTOVA ZASNOVANIH NA U-STATISTIKAMA I V-STATISTIKAMA SA NEDEGENERISANIM I SLABO DEGENERISANIM JEZGROM

Title: | ASIMPTOTSKA SVOJSTVA NEPARAMETARSKIH TESTOVA ZASNOVANIH NA U-STATISTIKAMA I V-STATISTIKAMA SA NEDEGENERISANIM I SLABO DEGENERISANIM JEZGROM |

Author: | Milošević, Bojana |

Abstract: | 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. |

URI: | http://hdl.handle.net/123456789/4470 |

Date: | 2016 |

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