S.V. Astashkin, Y.V. Solodyannikov. CONVOLUTIONAL EMPIRICAL PROCESSES AND STATISTIC TESTS RELATED WITH THEM. Samara: Publishing house "Samara University", 2000. - 102 p.

   In different practical investigations while analyzing statistic observations of random variables it is often possible and necessary to make a priory suppositions about their distribution law. The source of such suppositions can be different: the analysis of values physical nature, definite theoretical view on the studing things or simply the reasons of convenience of statistic methods application. So testing of such suppositions becomes the problem of topical interest.
Further we will confine the discussion to the case of continuous distributions and simple hypothesizes, i.e. the suppositions that this random variable has the fixed distribution. For their testing the so called goodness-of-fit tests are used; the tests of Kolmogorov-Smirnov and omega-squared type tests (see, for instance, the review about them in [1] ) are of the most practical importance among other tests. In both groups of tests the difference between empirical and theoretical distribution functions which is called an empirical process is of vital importance. Its asymptotic properties are discussed, for instance, in [2] (see also § 1 Chapter I of this article). The tests of Kolmogorov - Smirnov are based on the supremum of empirical process and the tests omega-squared are based on the integral of its square. Both tests are non-parametrical (i.e. they do not depend on the definite form of the distribution under testing), reliable against any alternative. In many cases they ensure the effective testing of hypothesises, but the problem of finding and study of the other tests remains topical.
In § 2 Chapter I we study the asymptotic properties of the difference between empirical and theoretical distribution functions convolutions (which will be called further the convolutional distribution process). It has been shown that for the equal and different samples, as well, by the corresponding standardization in the limit we have gaussian process with some covariance function. Then comparing it with Brownian bridge (the limit of standardized standard empirical processes sequence), we choose the definite "weight" function. If now we apply the functional, as in Kolmogorov-Smirnov tests or omega-squared, to the product of this function and convolutional empirical process, then we obtain the statistics on which new goodness-of-fit tests can be based. They are also non-parametrical, reliable against any alternative and, moreover, they have one advantage: on the same critical region on the straight line their significant level is larger than the classic one. At the end of Chapter II these questions are considered from the probalistic metric theory viewpoint, we also study the "smoothed" versions of convolutional empirical processes.
Together with the usual applications (see[1]) the tests based on the convolution have one the most suitable application. It is the situations when the random variable under consideration is the sum of the other random variables (for instance, the calculation of the total errors of measuring methods and means).
In conclusion, there are the results of calculations simulation and comparison of the statistics under consideration obtained on the personal computer.

CONTENTS

• INTRODUCTION
  
  
• Chapter 1. CONVOLUTIONAL EMPIRICAL PROCESSES
  1. Empirical distribution function and its properties
  2. Asymptotic conduct of the empirical distribution functions convolutions deviation from their expectations
     
     
• Chapter 2. STATISTIC TESTS RELATED WITH THE ESTIMATION OF THE DIVIATIONS OF THE EMPIRICAOL DISTRIBUTION FUNCTIONS CONVOLUTION FROM THEIR EXPECTATIONS
  3. The limiting distribution of the empirical convolutional processes functionals
  4. The estimation of the diviation by the comparison with classic statistics
  5. The estimation of the diviation using K.Fernique's theorem
  6. Convolutional random metrics and statistic tests connected with them
  7. The convolution of smoothed empirical distribution functions
     
     
• SUPPLEMENT
  1. The results of comparison of convolutional tests with Kolmogorov's ones
  2. The results of comparison of convolutional "smoothed" tests with Kolmogorov's ones
     
     
• REFERENCES

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