519.248 Methods for solving the problem of non-parametric testing of Lehmann's hypotheses when testing parallel systems

Timonin V. I. (Bauman Moscow State Technical University), Tyannikova N. D. (Bauman Moscow State Technical University)

NON-PARAMETRIC STATISTICS, LEHMANN'S HYPOTHESIS, KOLMOGOROV — SMIRNOV TYPE CRITERION, KAPLAN — MEIER ESTIMATE


doi: 10.18698/2309-3684-2018-1-98112


The article considers the problem of testing the Lehmann power hypothesis for two censored samples. The Kolmogorov — Smirnov type criterion based on a comparison of the Kaplan — Meier type estimates of the distribution functions for each censored sample is developed to test the power hypothesis. A method for calculating the exact statistics distributions is described on the basis of the model of particle random walk over an integer lattice. The probability values are calculated for a wide range of possible sample sizes. The convergence of this statistics distribution to the standard Kolmogorov — Smirnov distribution is proved provided that the hypothesis being tested is valid. The properties of a power parameter estimate obtained by minimizing statistics are investigated by statistical modeling methods.


[1] Gnedenko B.V., Belyaev Y.K., Soloviev A.D. Matematicheskie metody v teorii nadezhnosti. Osnovnye kharakteristiki nadezhnosti i ikh statisticheskiy analiz [Mathematical methods in reliability theory. The main characteristics of reliability and statistical analysis]. Moscow, Librokom Publ., 2013, 584 p.
[2] Gnedenko B.V. Voprosy matematicheskoy teorii nadezhnosti [Problems of mathematical theory of reliability]. Moscow, Radio i svyaz Publ., 1983, 376 p.
[3] Balakrishnan N., Tripathi R.C., Kannan N. Journal of Statistical Planning and Inference, 2010, no. 140, pp. 559–573.
[4] Bagdanovichus V., Kruopis J. Nikulin M.S. Nonparametric tests for censored data. London, ISTE Ltd Publ., 2011, 233 p.
[5] Balakrishnan N., Cramer E. The Art of Progressive Censoring. Applications to Reliability and Quality. New York, Springer Publ., 2014, 645 p.
[6] Kaplan E.L., Meier P. JAM Statistics Association, 1958, no. 53, pp. 57–481.
[7] Timonin V.I., Tyannikova N.D. Fizicheskie osnovy priborostroeniya – Physical Bases of Instrumentation, [in print].
[8] Bolshev L.N., Smirnov N.V. Tablitsy matematicheskoy statistiki [Mathematical Statistics Tables]. Мoscow, Nauka Publ., 1983, 416 p.
[9] Timonin V.I. Teoriya veroyatnostey i ee primenenie – Theory of Probability and its Applications, 1987, vol. 32, no. 4, pp. 790–792.
[10] Tyannikova N.D. Razrabotka neparametricheskih metodov analiza tsenzurirovannykh dannykh pri otsenke nadezhnosti slozhnykh tekhnicheskikh sistem v razlichnykh rezhimakh ispytaniy. Avtoreferat diss. cand. fiz.-mat. nauk [Development of nonparametric methods for analyzing censored data in assessing the reliability of complex technical systems in various test modes. Cand. phys. and math. sc. diss. Abstract]. Moscow, 2014, 16 p.
[11] Timonin V.I., Tyannikova N.D. Matematicheskoe modelirovanie i chislennye menody – Mathematical Modeling and Computational Methods, 2015, no. 3 (7), pp. 89–100.
[12] Hajek J., Sidak Z. Theory of rank tests. London, Academic Press Publ., 2004, 438 p.
[13] Sadykhov G.S, Krapotkin V.G., Kazakov O.I. Matematicheskoe modelirovanie i chislennye menody – Mathematical Modeling and Computational Methods, 2014, no. 1, pp. 82-98.


Тимонин В.И., Тянникова Н.Д. Методы решения задачи непараметрической проверки гипотез Лемана при испытаниях параллельных систем. Математическое моделирование и численные методы, 2018, № 1, с. 98-112



Download article

Колличество скачиваний: 31