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

• F. V. Motsnyi National Academy of Statistics, Accounting and Audit
Keywords: математична статистика, вибірка, випадкові величини, статистичні гіпотези, число ступенів вільності; критична точка; емпірична частота, теоретична частота, емпірична функція, теоретична функція, закони розподілу, непараметричні критерії, критерій Пірсона, критерій Колмогорова

### Abstract

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.

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