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Розглянуто статистичні критерії узгодження, що використовуються для перевірки гіпотез про закони розподілу генеральної сукупності. З єдиної позиції всебічно проаналізовані відомі непараметричні критерії Пірсона та Колмогорова. З’ясовані особливості їх застосування. Узагальнені результати численних теоретичних і прикладних досліджень. Запропоновані та розв’язані типові задачі. 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 signedrank
test and so on. Pearson and Kolmogorov criteria are most frequently used in mathematical statistics.
Pearson criterion (χ 2-criterion) is the universal statistical nonparametric criterion which has χ 2
-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.
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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