The thesis is devoted to the application of a new class of probability metrics, N-distances, introduced by Klebanov (Klebanov, 2005; Zinger et al., 1989), to the problems of verification of the classical statistical hypotheses of goodness of fit, homogeneity, symmetry and independence.

First of all a construction of statistics based on N-metrics for testing mentioned hypotheses is proposed. Then the problem of determination of the critical region of the criteria is investigated. The main results of the thesis are connected with the asymptotic behavior of test statistics under the null and alternative hypotheses. In general case the limit null distribution of proposed in the thesis tests statistics is established in terms of the distribution of infinite quadratic form of random normal variables with coefficients dependent on eigenvalues and functions of a certain integral operator. It is proved that under the alternative hypothesis the test statistics are asymptotically normal. In case of parametric hypothesis of goodness of fit particular attention is devoted to normality and exponentiality criteria. For hypothesis

of homogeneity a construction of multivariate distribution-free two-sample test is proposed. Testing the hypothesis of uniformity on hypersphere in more detail S1 and S2 cases are investigated.

In conclusion, a comparison of N-distance tests with some classical criteria is provided. For simple hypothesis of goodness of fit in univariate case as a measure

for comparison an Asymptotic Relative Efficiency (ARE) by Bahadur (Bahadur, 1960; Nikitin, 1995) is considered. In parallel to the theoretical results the empirical comparison of the power of the tests is examined by means of Monte Karlo simulations. Besides simple and composite hypotheses of goodness of fit, hypotheses

of uniformity on S1 and S2, we consider two-sample tests in uni- and multivariate cases. A wide range of alternative hypotheses are investigated.

Raktažodžiai: Goodness of fit, homogeneity, independence, symmetry tests, N-distance