Модель анализа совпадений при расчете непараметрических ранговых статистик

Abstract

Nonparametric statistic is one of the points of growth of modern mathematical and statistical methods of research. In nonparametric statistics, rank criteria based on the use of the ranks of the sample elements (observation results), rather than numerical values of the sample elements themselves, take an important place. Ranks are the numbers of sample elements in the corresponding variation series, constructed by rearranging the results of observations in the order of nondecreasing. Distributions of rank criteria are obtained on the assumption of continuity of the distribution functions of the observation results, hence, the probability of coincidence of the values of the random variables forming the analyzed samples should be equal to zero. However, in actual data, there are coincidences. Consequently, the assumption of the continuity of the distribution functions of the observation results is incorrect and known theorems on the distribution of rank statistics, strictly speaking, are not applicable. However, with a small number of coincidences, ranks statistics can be recommended for use, albeit with some corrections. Thus, an additional superstructure is mounted on the classical mathematical-statistical theory to take into account the coincidence of the data. Naturally, the validity of different methods used for accounting the coincidence of calculation data should be considered. We propose a probabilistic-statistical model that explains the occurrence of the coincidences and provides algorithms for their analysis. This model is based on the assumption that data coincidences appears as a result of «sticking together» of the slightly different observation results. We propose to introduce small corrections into each elements of the coincident group of observation results and thus to obtain a sample without coincidences and calculate the value of rank statistics. Having considered various variants of amendments, we obtain a «cloud» of values of rank statistics. Analysis of this «cloud» allows us to obtain statistical conclusions. Two-sample Wilcoxon test is considered as an example.