Verification of the gauss’s law with the unknown parameters by the Kolmogorov criterion in the problems of railways electric supply

Traction currents and voltages in the overhead railroad contact line hypothetically obey the Gauss’s distribution law. An algorithm for statistical modeling of the Kolmogorov distribution for the unknown values of the Gaussian law parameters has been developed. For a large number of the modeling program implementations, it has been shown that testing the hypothesis about the distribution law without taking into account the number of unknown parameters leads to an unacceptable increase in the criterion significance and a decrease in its power. The distribution of the Kolmogorov criterion is calculated with sufficient accuracy for two unknown parameters, which allows its wide practical use.

The laws of distribution of some parameters when organizing the movement of trains

The article discusses the key factors for the organization of train traffic. The analysis of statistical parameters of the intervals of departure of freight trains from railway stations and the length of the freight train flow is carried out. Based on these train schedules, graphical dependences of the intervals of departure of trains from separate points on the volume of train traffic and the distribution of train lengths in even and odd directions are constructed. Polynomial dependencies with coefficients of variation are derived. The curve of the second-order Erlang distribution function for the departure intervals is constructed. It is determined that the average departure intensity in one hour is equal to three trains, and the average departure interval lies between the 25th and 35th minutes with a probability of 0.793. There is a correlation between the coefficient of variation (mean square deviation) of the departure interval and its average daily deviation. The power function of the analytical ratio of the coefficient of variation from the average interval of departure of freight trains from the technical stations of the Krasnoyarsk railway during the day is determined using the least squares method. The degrees of correspondence of theoretical and statistical distributions of the number of cars in trains are studied using the Kolmogorov criterion with the calculation of the probability value and confirmation that the distribution of the number of cars in trains obeys the binomial law. Using the Pearson agreement criterion, it is proved that the frequency of departure of freight trains from railway stations obeys the exponential distribution law.

Determination of Boundary Parameters of Pipelines Elbows Blanking Elements Made from Titanium Alloy Ti-4Al-2Mn and Steel AISI 304

The results of computer simulation processing and determination of limiting degree of deformation by the Kolmogorov criterion are presented. The derivation of the dependence for the evaluation of the destruction of low-plastic materials at the edge is made at bend angle of 900

Analysis of Nonparametric and Parametric Criteria for Statistical Hypotheses Testing. Chapter 1. Agreement Criteria of Pearson and Kolmogorov

In the statistical analysis of experimental results it is extremely important to know the distribution laws of the general population. ‎Because of all assumptions about the distribution laws are statistical hypotheses, they should be tested. ‎Testing hypotheses are carried out by using the statistical criteria that divided the multitude in two subsets: null and alternative. The ‎null hypothesis is accepted in subset null and is rejected in alternative subset. Knowledge of the distribution law is a prerequisite for the use of numerical mathematical methods. The hypothesis is accepted if the divergence between empirical and theoretical distributions will be random. The hypothesis is rejected if the divergence between empirical and theoretical distributions will be essential. There is a number of different agreement criteria for the statistical hypotheses testing. The paper continues ideas of the author’s works, devoted to advanced based tools of the mathematical statistics. This part of the paper is devoted to nonparametric agreement criteria. Nonparametric tests don’t allow us to include in calculations the parameters of the probability distribution and to operate with frequency only, as well as to assume directly that the experimental data have a specific distribution. Nonparametric criteria are widely used in analysis of the empirical data, in the testing of the simple and complex statistical hypotheses etc. They include the well known criteria of K. Pearson, A. Kolmogorov, N. H. Kuiper, G. S. Watson, T. W. Anderson, D. A. Darling, J. Zhang, Mann – Whitney U-test, Wilcoxon signed-rank test and so on. Pearson and Kolmogorov criteria are most frequently used in mathematical statistics. Pearson criterion (-criterion) is the universal statistical nonparametric criterion which has -distribution. It is used for the testing of the null hypothesis about subordination of the distribution of sample empirical to theory of general population at large amounts of sample (n>50). Pearson criterion is connected with calculation of theoretical frequency. Kolmogorov criterion is used for comparing empirical and theoretical distributions and permits to find the point in which the difference between these distributions is maximum and statistically reliable. Kolmogorov criterion is used at large amounts of sample too. It should be noted, that the results obtained by using Pearson criterion are more precise because practically all experimental data are used. The peculiarities of Pearson and Kolmogorov criteria are found out. The formulas for calculations are given and the typical tasks are suggested and solved. The typical tasks are suggested and solved that help us to understand more deeply the essence of Pearson and Kolmogorov criteria.

Reversible Markov processes on general spaces and spatial migration processes

In this study, we characterize the equilibrium behavior of spatial migration processes that represent population migrations, or birth-death processes, in general spaces. These processes are reversible Markov jump processes on measure spaces. As a precursor, we present fundamental properties of reversible Markov jump processes on general spaces. A major result is a canonical formula for the stationary distribution of a reversible process. This involves the characterization of two-way communication in transitions, using certain Radon-Nikodým derivatives. Other results concern a Kolmogorov criterion for reversibility, time reversibility, and several methods of constructing or identifying reversible processes.

Reversible Markov processes on general spaces and spatial migration processes

In this study, we characterize the equilibrium behavior of spatial migration processes that represent population migrations, or birth-death processes, in general spaces. These processes are reversible Markov jump processes on measure spaces. As a precursor, we present fundamental properties of reversible Markov jump processes on general spaces. A major result is a canonical formula for the stationary distribution of a reversible process. This involves the characterization of two-way communication in transitions, using certain Radon-Nikodým derivatives. Other results concern a Kolmogorov criterion for reversibility, time reversibility, and several methods of constructing or identifying reversible processes.

On asymptotics of a class of 1-dimensional diffusions

In this paper we study the almost sure asymptotics of htXt where (Xt) satisfies a stochastic differential equation and ht is a deterministic differentiable function, ht ↓ 0 as t → ∞. We give necessary and sufficient criteria, in terms of deterministic integrals, for the process htXt to converge to zero, to oscillate boundedly and for the case of infinite oscillations. Necessary and sufficient criteria for convergence to zero, in terms of a related integral, was studied by Chan and Williams in [1], for a class of diffusions that arise in simulated annealing. In Chan [2] the results are extended to higher dimensions. For Brownian motion, Jeulin and Yor[3] give a general criterion for convergence to zero, when the function ht is merely continuous and non-negative. The results of [1] and [3] are both generalizations of the well known ‘Kolmogorov criterion’ for Brownian motion. Our approach is very close to the techniques developed by Chan and Williams in [1]. Indeed, our method is essentially a fine tuning of those techniques. We apply our results, in Section 3, to a class of diffusions and compute the critical rate ht for which bounded oscillations occur. For Brownian motion this is the law of iterated logarithm, modulo the size of the oscillation. We also consider two parametrized families of diffusions for which the critical rate is seen to depend smoothly on the parameter. These results appear to be new. Our method also gives estimates on the size of the oscillations (Section 6).