The Uniform Variants of the Glivenko-Cantelli and Donsker Type Theorems for a Sequential Integral Process of Independence

Abdushukurov Abdurahim Ahmedovich; Kakadjanova Leyla Reshitovna

doi:doi:10.11648/j.ajtas.20200904.15

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The Uniform Variants of the Glivenko-Cantelli and Donsker Type Theorems for a Sequential Integral Process of Independence

Abdushukurov Abdurahim Ahmedovich, Kakadjanova Leyla Reshitovna

Published in American Journal of Theoretical and Applied Statistics (Volume 9, Issue 4)

Received: 15 February 2020 Accepted: 22 May 2020 Published: 4 June 2020

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Abstract

In the analysis of statistical data in biomedical treatments, engineering, insurance, demography, and also in other areas of practical researches, the random variables of interest take their possible values depending on the implementation of certain events. So in tests of physical systems (or individuals) on duration of uptime values of operating systems depend on subsystems failures, in insurance business insurance company payments to its customers depend on insurance claims. In such experimental situations, naturally become problems of studying the dependence of random variables on the corresponding events. The main task of statistics of such incomplete observations is estimating the distribution function or what is the same, the survival function of the tested objects. To date, there are numerous estimates of these characteristics or their functionals in various models of incomplete observations. In this paper investigated the asymptotic properties of sequential processes of independence of the integral structure and uniform versions of the strong law of large numbers and the central limit theorem for integral processes of independence by indexed classes are established. The obtained results can be used to construct statistics of criteria for testing a hypothesis of independence of random variables on the corresponding events.

Published in	American Journal of Theoretical and Applied Statistics (Volume 9, Issue 4)
DOI	10.11648/j.ajtas.20200904.15
Page(s)	121-126
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Empirical Processes, Metric Entropy, Glivenko-Cantelli and Donsker Theorems

References

[1]	Abdushukurov A. A., Kakadjanova L. R. Empirical processes of independence indexed by class of measurable functions. Acta NUUz., vol. 1. issue 2/1 pp. 15-20. 2014. (In Russian).
[2]	Abdushukurov A. A., Kakadjanova L. R. A class of special empirical process of independence. J. Siberian Federal Univ. Math. Phys, vol. 8. issue 2, pp. 125-133, 2015.
[3]	Abdushukurov A. A., Kakadjanova L. R. Sequential empirical process of independence. J. Siberian Federal Univ. Math. Phys., vol. 11. issue 5, pp. 634-643, 2018.
[4]	Bae J., Kim S. The sequential uniform law of large numbers. Bull. Korean Math. Soc., vol. 43, issue 3, pp. 479-486, 2006.
[5]	Dudley R. M. Uniform central limit theorems. Cambridge University Press, Cambridge, 1999.
[6]	Dudley R. M. Notes on empirical process. Cambridge. January 24. 2000.
[7]	Kakadjanova L. R. Empirical process of independence in presence of estimated parameter. Proceedings of the International Workshop “Applied Methods of Statistical Analysis”, AMSA-2019. Novosibirsk. Russia. pp. 96-102, 18-19 September, 2019.
[8]	Mason D. M. Classical empirical process theory and weighted approximations. Cominicaciones del CIMAT. 1-15-03/09-11-2015 (PE/CIMAT).
[9]	Mason D. M. Selected defenitions and results from modern empirical process theory. Comunicaciones del CIMAT. 1-17-01, 16.03.2017 (PE).
[10]	Ossiander Mina. A central limit theorem under metric entropy with L2 bracketing. Ann. Probab., vol. 15, issue 3, pp. 897-919, 1987.
[11]	Prokhorov Yu. An enlarge of S. N. Bernstein's inequality to the multivariate case. Theory Probab. Appl., vol. 13, issue 3, pp. 266-274, 1968. (In Russian).
[12]	Van der Vaart A. W., Wellner J. A. Weak convergence and empirical processes. Springer. 1996.
[13]	Van der Vaart A. W. Asymptotic Statistics. Cambridge University Press. Cambridge. 1998.
[14]	Vapnik V. N. Statistical learning theory. Wiley, New York, 1998.
[15]	Varron D. Donsker and Glivenko-Cantelly theorems for a class of processes generalizing the empirical process. Electronical J. Statist., 8, pp. 2296-2320, 2014.

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Abdushukurov Abdurahim Ahmedovich, Kakadjanova Leyla Reshitovna. (2020). The Uniform Variants of the Glivenko-Cantelli and Donsker Type Theorems for a Sequential Integral Process of Independence. American Journal of Theoretical and Applied Statistics, 9(4), 121-126. https://doi.org/10.11648/j.ajtas.20200904.15

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Abdushukurov Abdurahim Ahmedovich; Kakadjanova Leyla Reshitovna. The Uniform Variants of the Glivenko-Cantelli and Donsker Type Theorems for a Sequential Integral Process of Independence. Am. J. Theor. Appl. Stat. 2020, 9(4), 121-126. doi: 10.11648/j.ajtas.20200904.15

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AMA Style

Abdushukurov Abdurahim Ahmedovich, Kakadjanova Leyla Reshitovna. The Uniform Variants of the Glivenko-Cantelli and Donsker Type Theorems for a Sequential Integral Process of Independence. Am J Theor Appl Stat. 2020;9(4):121-126. doi: 10.11648/j.ajtas.20200904.15

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@article{10.11648/j.ajtas.20200904.15,
  author = {Abdushukurov Abdurahim Ahmedovich and Kakadjanova Leyla Reshitovna},
  title = {The Uniform Variants of the Glivenko-Cantelli and Donsker Type Theorems for a Sequential Integral Process of Independence},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {9},
  number = {4},
  pages = {121-126},
  doi = {10.11648/j.ajtas.20200904.15},
  url = {https://doi.org/10.11648/j.ajtas.20200904.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20200904.15},
  abstract = {In the analysis of statistical data in biomedical treatments, engineering, insurance, demography, and also in other areas of practical researches, the random variables of interest take their possible values depending on the implementation of certain events. So in tests of physical systems (or individuals) on duration of uptime values of operating systems depend on subsystems failures, in insurance business insurance company payments to its customers depend on insurance claims. In such experimental situations, naturally become problems of studying the dependence of random variables on the corresponding events. The main task of statistics of such incomplete observations is estimating the distribution function or what is the same, the survival function of the tested objects. To date, there are numerous estimates of these characteristics or their functionals in various models of incomplete observations. In this paper investigated the asymptotic properties of sequential processes of independence of the integral structure and uniform versions of the strong law of large numbers and the central limit theorem for integral processes of independence by indexed classes are established. The obtained results can be used to construct statistics of criteria for testing a hypothesis of independence of random variables on the corresponding events.},
 year = {2020}
}

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TY  - JOUR
T1  - The Uniform Variants of the Glivenko-Cantelli and Donsker Type Theorems for a Sequential Integral Process of Independence
AU  - Abdushukurov Abdurahim Ahmedovich
AU  - Kakadjanova Leyla Reshitovna
Y1  - 2020/06/04
PY  - 2020
N1  - https://doi.org/10.11648/j.ajtas.20200904.15
DO  - 10.11648/j.ajtas.20200904.15
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 121
EP  - 126
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20200904.15
AB  - In the analysis of statistical data in biomedical treatments, engineering, insurance, demography, and also in other areas of practical researches, the random variables of interest take their possible values depending on the implementation of certain events. So in tests of physical systems (or individuals) on duration of uptime values of operating systems depend on subsystems failures, in insurance business insurance company payments to its customers depend on insurance claims. In such experimental situations, naturally become problems of studying the dependence of random variables on the corresponding events. The main task of statistics of such incomplete observations is estimating the distribution function or what is the same, the survival function of the tested objects. To date, there are numerous estimates of these characteristics or their functionals in various models of incomplete observations. In this paper investigated the asymptotic properties of sequential processes of independence of the integral structure and uniform versions of the strong law of large numbers and the central limit theorem for integral processes of independence by indexed classes are established. The obtained results can be used to construct statistics of criteria for testing a hypothesis of independence of random variables on the corresponding events.
VL  - 9
IS  - 4
ER  -

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Author Information

Abdushukurov Abdurahim Ahmedovich

Department of Applied Mathematics and Informatics, Faculty of Applied Mathematics and Informatics, Tashkent Branch of Moscow State University Named After M. V. Lomonosov, Tashkent, Uzbekistan
Kakadjanova Leyla Reshitovna

Department of Probability Theory and Mathematical Statistics, Faculty of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan

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APA Style

Abdushukurov Abdurahim Ahmedovich, Kakadjanova Leyla Reshitovna. (2020). The Uniform Variants of the Glivenko-Cantelli and Donsker Type Theorems for a Sequential Integral Process of Independence. American Journal of Theoretical and Applied Statistics, 9(4), 121-126. https://doi.org/10.11648/j.ajtas.20200904.15

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ACS Style

Abdushukurov Abdurahim Ahmedovich; Kakadjanova Leyla Reshitovna. The Uniform Variants of the Glivenko-Cantelli and Donsker Type Theorems for a Sequential Integral Process of Independence. Am. J. Theor. Appl. Stat. 2020, 9(4), 121-126. doi: 10.11648/j.ajtas.20200904.15

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AMA Style

Abdushukurov Abdurahim Ahmedovich, Kakadjanova Leyla Reshitovna. The Uniform Variants of the Glivenko-Cantelli and Donsker Type Theorems for a Sequential Integral Process of Independence. Am J Theor Appl Stat. 2020;9(4):121-126. doi: 10.11648/j.ajtas.20200904.15

Copy | Download

@article{10.11648/j.ajtas.20200904.15,
  author = {Abdushukurov Abdurahim Ahmedovich and Kakadjanova Leyla Reshitovna},
  title = {The Uniform Variants of the Glivenko-Cantelli and Donsker Type Theorems for a Sequential Integral Process of Independence},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {9},
  number = {4},
  pages = {121-126},
  doi = {10.11648/j.ajtas.20200904.15},
  url = {https://doi.org/10.11648/j.ajtas.20200904.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20200904.15},
  abstract = {In the analysis of statistical data in biomedical treatments, engineering, insurance, demography, and also in other areas of practical researches, the random variables of interest take their possible values depending on the implementation of certain events. So in tests of physical systems (or individuals) on duration of uptime values of operating systems depend on subsystems failures, in insurance business insurance company payments to its customers depend on insurance claims. In such experimental situations, naturally become problems of studying the dependence of random variables on the corresponding events. The main task of statistics of such incomplete observations is estimating the distribution function or what is the same, the survival function of the tested objects. To date, there are numerous estimates of these characteristics or their functionals in various models of incomplete observations. In this paper investigated the asymptotic properties of sequential processes of independence of the integral structure and uniform versions of the strong law of large numbers and the central limit theorem for integral processes of independence by indexed classes are established. The obtained results can be used to construct statistics of criteria for testing a hypothesis of independence of random variables on the corresponding events.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - The Uniform Variants of the Glivenko-Cantelli and Donsker Type Theorems for a Sequential Integral Process of Independence
AU  - Abdushukurov Abdurahim Ahmedovich
AU  - Kakadjanova Leyla Reshitovna
Y1  - 2020/06/04
PY  - 2020
N1  - https://doi.org/10.11648/j.ajtas.20200904.15
DO  - 10.11648/j.ajtas.20200904.15
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 121
EP  - 126
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20200904.15
AB  - In the analysis of statistical data in biomedical treatments, engineering, insurance, demography, and also in other areas of practical researches, the random variables of interest take their possible values depending on the implementation of certain events. So in tests of physical systems (or individuals) on duration of uptime values of operating systems depend on subsystems failures, in insurance business insurance company payments to its customers depend on insurance claims. In such experimental situations, naturally become problems of studying the dependence of random variables on the corresponding events. The main task of statistics of such incomplete observations is estimating the distribution function or what is the same, the survival function of the tested objects. To date, there are numerous estimates of these characteristics or their functionals in various models of incomplete observations. In this paper investigated the asymptotic properties of sequential processes of independence of the integral structure and uniform versions of the strong law of large numbers and the central limit theorem for integral processes of independence by indexed classes are established. The obtained results can be used to construct statistics of criteria for testing a hypothesis of independence of random variables on the corresponding events.
VL  - 9
IS  - 4
ER  -

Copy | Download