Power Analysis of Two Kolmogorov --- Smirnov Type Criteria in Testing the Cox Power Model for the Progressively Censored Samples

Authors: Timonin V.I., Tiannikova N.D. Published: 11.03.2024
Published in issue: #2(113)/2024  

Category: Mathematics and Mechanics | Chapter: Mathematical Simulation, Numerical Methods and Software Packages  
Keywords: Kolmogorov --- Smirnov criterion, Cox model, Kaplan --- Meier estimate, criterion power, sequential systems, censoring


Testing a technical system often leads to a problem of comparing reliability indicators of their elements in different modes (operation in different climatic zones, comparing results of laboratory tests and real operation data, etc.). At the same time, laws of the element time to failure distribution are unknown. Nature of the available data causes additional difficulties in solving this problem, i.e. a failure of one element in a sequential system leads to the fact that time to failure of the remaining usable elements remains unknown (censored). In this regard, it becomes relevant to solve such a problem using the nonparametric (free from knowing distribution) methods. Previously, the authors developed a Kolmogorov --- Smirnov type criterion making it possible to check power dependence of the reliability functions with elements of the sequential systems (nonparametric Cox power model). Statistics of this criterion uses the Kaplan --- Meier estimates of the elements reliability functions constructed from developments in systems composed of them. However, a feature of the Kaplan --- Meier estimates is their slow convergence to the theoretical reliability function on the distribution right tail, which could lead to a decrease in the criterion power (decrease in sensitivity). Here, another criterion of the Kolmogorov --- Smirnov type is proposed to solve a similar problem, its statistics is not using the Kaplan --- Meier estimates and is based on comparing estimates of the system reliability functions. Exact and asymptotic distributions of its statistics are obtained. Detailed study of the two criteria power is carried out using numerical analysis and statistical simulation. The Cox parameter estimates accuracy is obtained by minimizing the compared criteria statistics and is analyzed using the Monte Carlo methods

Please cite this article in English as:

Timonin V.I., Tiannikova N.D. Power analysis of two Kolmogorov --- Smirnov type criteria in testing the Cox power model for the progressively censored samples. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2024, no. 2 (113), pp. 57--73 (in Russ.). EDN: KDBJJP


[1] Gnedenko B.V., Belyaev Yu.K., Solovyev A.D. Matematicheskie metody v teorii nadezhnosti [Mathematical methods in reliability theory]. Moscow, Librokom Publ., 2012.

[2] Abdushukurov A.A. Estimation of survival function in cox regression model under random censoring from both sides. Commun. Stat. Theory Methods, 2015, vol. 44, iss. 3, pp. 533--553. DOI: https://doi.org/10.1080/03610926.2012.746981

[3] Nikulin M., Wu H.I. The cox model and its applications. Berlin, Springer, 2016.

[4] Escobar L.A., Meeker W.Q. A review of accelerated test models. Statist. Sci., 2006, vol. 21, iss. 4, pp. 552--577. DOI: https://doi.org/10.1214/088342306000000321

[5] Balakrishnan N., Tripathi R.C., Kannan N., et al. Some nonparametric precedence type tests based on progressively censored samples and evaluation of power. J. Stat. Plan. Inference, 2010, vol. 140, iss. 2, pp. 559--573. DOI: https://doi.org/10.1016/j.jspi.2009.08.003

[6] Dimitrova D.S., Kaishev V.K., Tan S. Computing the Kolmogorov --- Smirnov distribution when the underlying CDF is purely discrete, mixed or continuous. J. Stat. Softw., 2020, vol. 95, iss. 10, pp. 1--42. DOI: https://doi.org/10.18637/jss.v095.i10

[7] Timonin V.I., Tyannikova N.D. Application of Kaplan --- Meier estimates to testing Cox power hypothesis for two progressive lycensored samples. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2015, no. 6 (63), pp. 68--84 (in Russ.). DOI: http://dx.doi.org/10.18698/1812-3368-2015-6-68-84

[8] Ng N., Balakrishnan N. Precedence-type test based on Kaplan --- Meier estimator of cumulative distribution function. J. Stat. Plan. Inference, 2010, vol. 140, iss. 8, pp. 2295--2311. DOI: https://doi.org/10.1016/j.jspi.2010.01.025

[9] Su P., Li C., Shyr Y. Sample size determination for paired right-censored data based on the difference of Kaplan --- Meier estimates. Comput. Stat. Data Anal., 2014, vol. 74, pp. 39--51. DOI: https://doi.org/10.1016/j.csda.2013.12.006

[10] Bagdonavichus V., Kruopis J. Nikulin M.S. Nonparametric tests for censored data. London, ISTE Ltd, 2011.

[11] Balakrishnan N., Cramer E. The art of progressive censoring. New York, Springer, 2014.

[12] Maturi T.A., Coolen-Schrijner P., Coolen F.P. Nonparametric predictive comparison of lifetime data under progressive censoring. J. Stat. Plan. Inference, 2010, vol. 140, iss. 2, pp. 515--525. DOI: https://doi.org/10.1016/j.jspi.2009.07.027

[13] Timonin V.I., Tyannikova N.D. The method of calculating the exact distributions of the Kolmogorov --- Smirnov statistics in case of violation of homogeneity and independence of the analyzed samples. Nauka i obrazovanie: nauchnoe izdanie [Science and Education: Scientific Publication], 2014, no. 11 (in Russ.). DOI: 10.7463/1114.0740251

[14] Simard R., L’Ecuyer P. Computing the two-sided Kolmogorov --- Smirnov distribution. J. Stat. Softw., 2011, vol. 39, iss. 11, pp. 1--18. DOI: https://doi.org/10.18637/jss.v039.i11

[15] Lebedev A.V., Fadeeva L.N. Teoriya veroyatnostey i matematicheskaya statistika [Theory of probability and mathematical statistics]. Moscow, EKSMO Publ., 2010.

[16] Lemeshko B., Veretel’nikova I. Power of k-sample tests aimed at checking the homogeneity of laws. Meas. Tech., 2018, vol. 61, no. 2, pp. 647--654. DOI: https://doi.org/10.1007/s11018-018-1479-1

[17] Hajek J., Sidak Z. Theory of rank tests. London, Academic Press, 1999.

[18] Borzykh D.A. On a class of functionals continuous in the Skorokhod topology. Vestnik Voronezhskogo gosudarstvennogo universiteta. Ser. Fizika. Matematika [Proceedings of Voronezh State University. Ser. Physics. Mathematics], 2016,no. 4, pp. 83--88 (in Russ.). EDN: WYMNWL

[19] Han D., Bai T. On the maximum likelihood estimation for progressively censored lifetimes from constant-stress and step-stress accelerated tests. Electron. J. Appl. Stat. Anal., 2019, vol. 12, no. 2, pp. 392--404. DOI: https://doi.org/10.1285/i20705948v12n2p392

[20] Kroese D.P., Brereton T., Taimre T., et al. Why the Monte Carlo method is so important today. WIREs Comput Stat., 2014, vol. 6, iss. 6, pp. 386--392. DOI: https://doi.org/10.1002/wics.1314