A study of smallsample estimates of distribution parameters
https://doi.org/10.21683/1729-2646-2025-25-4-29-35
Abstract
Aim. Evaluating distribution parameters based on small samples is an unconventional problem in itself. Solving it by maximizing the likelihood function may produce highly biased results. The paper analyses the properties of some small-sample estimates of beta distribution parameters of the 1-st kind. Methods. The comparison of small-sample estimates of beta distribution parameters using various methods involved simulation with the number of tests N = 104. Results. Parameter estimation using the maximum likelihood method does produce a highly biased result for small samples. The bootstrap method, as compared to the maximum likelihood method, produces less biased estimates with a smaller variance. The most acceptable (close to the initial values) result was obtained using the mathematical expectation (or median) and variance. Conclusion. For small samples, no particular method of parametre estimation can be recommended. The neural network analysis appears to be the best suited for small samples. Neural network integration of a number of methods of estimation may significantly improve its accuracy.
About the Author
A. V. Volovik
JSC “UEC-Klimov”
Россия
Россия
Alexander V. Volovik, Candidate of Engineering, Lead Design Engineer
16 bldg. А, apt. 128, Tsentralnaya, Shushary, ter. Detskoselsky, Saint Petersburg, 196634
References
1. Volovik A.V. A combinatorial method of small sample identification. Dependability 2024;24(2):3-7. (in Russ.). DOI: 10.21683/1729-2646-2024-24-2-3-7
2. Johnson N., Leone F. Statistics and Experimental Designs and Engineering and the Physical Sciences. Mathods of Data Processing. Moscow: Mir; 1980.
3. Gaskarov D.V., Shapovalov V.I. [Small sample]. Moscow: Statistika; 1978. (in Russ.)
4. Orlov A.I. [Econometrics]. Moscow: Examen; 2002. (in Russ.)
5. Popukailo V.S. [Decision support for small-size passive samples]. A Doctor of Computer Science dissertation. UDC 004.415.2. Chisinau; 2017. (in Russ.)
6. Johnson N.L. Continuous Univariate Distributions: in 2 volumes. Volume 2. Moscow: BINOM. Laboratoriya znaniy; 2010-2012.
7. Lemeshko B.Yu., Blinov P.Yu [Criteria for testing a distribution for deviation from a uniform law. An application guide]. Novosibirsk: NSTU; 2015. (in Russ.)
8. Korn G., Korn T. Mathematical handbook. Moscow: Nauka; 1974.
9. Vadzinsky R.N. Handbook of Probability Distributions. St. Petersburg: Nauka; 2001. (in Russ.)
10. Gromov Yu.Yu., Karpov I.G. [Distribution laws of a continuous random variable with maximum entropy. Generalized method of moments]. Nauchno-tekhnicheskie vedomosti SPBGPU 1 2009:37-41. (in Russ.)
11. Rebrova O. [Average or median though]. Troitsky variant 2011;90:13. (in Russ.)
12. GOST R 50779.24-2005 (ISO 8595:1990): Interpretation of statistical data. Estimation of a median. Мoscow: Standartinform; 2005. (in Russ.)
13. Kerman J. A closed-form approximation for the median of the beta distribution. (accessed 29.9.2025). Available at: https://arxiv.org/abs/1111.0433.
14. Lemeshko B.Yu., Hildebrant S.Ya., Postovalov S.N. [On the assessment of reliability parameters based on censored samples]. Industrial Laboratory. Diagnostics of Materials 2001;67(1):52-64. (in Russ.)
15. Volchikhin V.I. et al. The neural network analysis of normality of small samples of biometric data through using the Chi-square test and Anderson–Darling criteria. Inzhenernyye tekhnologii i sistemy 2019;29(2):205-217. (in Russ.)
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For citations:
Volovik A.V. A study of smallsample estimates of distribution parameters. Dependability. 2025;25(4):29-37. (In Russ.) https://doi.org/10.21683/1729-2646-2025-25-4-29-35
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ISSN 1729-2646 (Print)
ISSN 2500-3909 (Online)
ISSN 2500-3909 (Online)
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