Open AccessTo establish the relationship between the distribution functions of the experimental results for different values of the external factors are most commonly used parametric models in which the parameters of the distribution functions depend on factors, and their views do not change. Meanwhile, when we have a small amount of data (and this is more common in practice), the distribution function is often unknown, and it is difficult to determine. Hence, it is of great importance to evaluate different relationships between the distribution laws without specifying a particular form of the distribution (these issues are handled by non-parametric statistics). The most common model, used to establish the relationship between the theoretical distribution functions of different samples, is the Cox model. Furthermore, even for testing the homogeneity of multiple samples, experiments, which are necessary to obtain them, are so complex that the obtained samples are dependent. So, all of these tasks requires the development of new nonparametric tests for dependency. Due to the fact, that the volume of the sample is always small, knowledge of exact distributions of statistics, which are used, is of special importance. The paper develops a general method for tabulating the exact distributions (for finite volumes of samples) of a wide class of statistics of the Kolmogorov-Smirnov test. With the appropriate specialization of the proposed algorithm, it allows us to calculate the distribution of various statistics of the specified type. In particular, it is applicable for calculating the distribution of statistics such as Kiefer-Gikhman used to check the dependencies between Lehmann distribution functions of several samples. With small modifications it allows us to tabulate the distribution statistics of the Kolmogorov-Smirnov used for checking the homogeneity of dependent samples. Along with the fact that the method has great generality, it also allows us to calculate the exact distribution for very large volumes of samples. This fact allows us to estimate the volume of the sample, in which the asymptotic distribution can be applied.
The limits of this method applicability are also given. It assumes the validity of a special model of random movement of particle on a multidimensional lattice in which the future behavior of the particle trajectory at presently given is independent of its past.