Estimation of gamma distribution parameters

Statements of problems of statistical analysis of data with a gamma distribution are related to classical mathematical statistics. Oddly enough, not all alone were solved within the framework of parametric statistics, which was at the forefront of the development of statistical science in the first third of the 20th century. As with the beta distribution, gaps need to be filled. This is necessary because the gamma distribution is currently widely used in theoretical and applied work. An example is GOST 11.011–83 «Applied statistics. Rules for determining estimates and confidence limits for gamma distribution parameters». The standard gamma distribution is determined by the shape parameter. When switching to a scale-shift family, scale and translation parameters are added. Seven formulations of parameter estimation problems are considered, since each of the three parameters can be either unknown or known. For each of the formulations, the estimates of the method of moments and their asymptotic variances are found. For a known shift parameter, maximum likelihood estimates are obtained. One-step estimates, asymptotically equivalent to maximum likelihood estimates, are used for an unknown shift parameter. The presence of measurement errors affects the accuracy of parameter estimates when applying certain calculation algorithms. In GOST 11.011–83, based on the interval data model, rules are given for choosing an estimation method for unknown shape and scale parameters and a known shift parameter. During the development of GOST 11.011–83, problems were identified, for the solution of which new methods from a scientific point of view were proposed. Further development of new scientific results obtained in the course of solving a practical problem (development of GOST 11.011–83) led to the creation of new scientific directions. We are talking about the statistics of interval data, as well as one-step estimates. To date, the statistics of interval data as a branch of mathematical statistics is quite developed and covers all the main areas of statistical methods. It is an important part of systemic fuzzy interval mathematics.

1. Balakrishnan N., Nevzorov V. B. A primer of statistical distributions. — New Jersey: Wiley-Interscience, 2003. — 328 p.

2. Hastings N., Peekock J. Handbook of Statistical Distributions. — Moscow: Statistika, 1980. — 95 p. [Russian translation].

3. Probability and mathematical statistics: Encyclopedia / Prokhorov Yu. V., Ed. — Moscow: Bol’shaya Ros. Entsikl., 1999. — 910 p. [in Russian].

4. Orlov A. I. Certification and statistical methods (summarizing article) / Zavod. Lab. Diagn. Mater. 1997. Vol. 63. N 3. P. 55 – 62 [in Russian].

5. Orlov A. I. Statistical analysis of samples from the beta distribution / Nauch. Zh. KubGAU. 2023. N 03(187). P. 184 – 206 [in Russian].

6. Ivshin V. V. Statistical problems of reliability estimation in the load-strength model in the cases of gamma and Weibull distributions / Vestn. Perm. Gos. Tekhn. Univ. Matem. Model. Sist. Prots. 1994. N 2. P. 43 – 49 [in Russian].

7. Ryazanskii V. P., Yudin S. V. Gamma function as the basis of a three-parameter distribution of parameters of accuracy and reliability of defense industry products / Izv. Tul. Gos. Univ. Tekhn. Nauki. 2022. N 2. P. 603 – 612 [in Russian].

8. Johnson N. L., Kotz S., Balakrishnan N. Continuous univariate distributions. Vol. 1. Second edition. — New York – Chichester – Brisbane – Toronto – Singapore: Jonh Wiley and Sons, 1994. — 777 p.

9. Benderskii A. M., Nevel’son M. B. Estimating the Shape Parameter of the Gamma Distribution / Nadezhn. Kontrol’ Kach. 1981. N 10. P. 14 – 21 [in Russian].

10. Shor Ya. B. Statistical methods of analysis and quality and reliability control. — Moscow: Sovetskoe radio, 1962. — 553 p. [in Russian].

11. Burgin T. A. The gamma distribution and inventory control / Operational Research Quarterly. 1975. Vol. 26. N 4. P. 377 – 519.

12. Svichinskii S. V., Makarichev A. V. Justification for the gamma distribution of route lengths for passengers traveling by route transport / Modern. Nauch. Issl. Transp. Kompl. 2013. Vol. 2. P. 341 – 349 [in Russian].

13. Volk A. M. Analysis of the properties of statistical estimates of the parameters of the generalized gamma distribution / Tr. BGTU. Ser. 3: Fiz.-Mat. Nauki Inform. 2023. N 1(266). P. 10 – 14 [in Russian].

14. Kudryavtsev A. A., Shestakov O. V., Shorgin V. S. Method for estimating the bending parameters, shape and scale of the gamma exponential distribution / Inform. Primen. 2021. Vol. 15. N 3. P. 57 – 62 [in Russian].

15. Karpov I. G., Zyryanov Yu. T., Mel’nik O. V. Model of the distribution law of continuous random variables based on the gamma distribution / Fund. Prikl. Probl. Tekhn. Tekhnol. 2014. N 3(305). P. 26 – 30 [in Russian].

16. Dolgov A. Yu., Tereshchenko E. V., Sorochan A. A. Study of a statistical hypothesis under the conditions of a gamma distribution / Vestn. Pridnestr. Univ. Ser. Fiz.-Mat. Tekhn. Nauki. Ékon. Upravl. 2018. N 3(60). P. 112 – 118 [in Russian].

17. Orlov A. I. Applied statistical analysis: textbook. — Moscow: IPR Media, 2022. — 812 p. [in Russian].

18. Yanke E., Emde F., Lesh F. Special functions. — Moscow: Nauka, 1964. — 344 p. [in Russian].

19. Kendall M. J., Stewart A. Statistical inferences and relationship. — Moscow: Nauka, 1973. — 896 p. [Russian translation].

20. Satarov G. A., Shmerling D. S. New statistical model for paired comparisons / Expert assessments in management problems: Collection of papers. — Moscow: Izd. IPU AN SSSR, 1982. P. 67 – 79 [in Russian].

21. Lapiga A. G. Multicriteria quality management problems: constructing a quality forecast on a point scale / Zavod. Lab. 1983. Vol. 49. N 7. P. 55 – 59 [in Russian].

22. Zaks Sh. The theory of statistical inference. — Moscow: Mir, 1975. — 776 p. [Russian translation].

23. Bakhmutov V. O., Kosarev L. N. Using the maximum likelihood method to evaluate the homogeneity of fatigue test results / Zavod. Lab. 1986. Vol. 52. N 5. P. 52 – 57 [in Russian].

24. Reznikova A. Ya., Shmerling D. S. Estimating the parameters of probabilistic models of paired and multiple comparisons / Statistical methods of estimation and testing of hypotheses: Interuniversity collection of scientific papers. — Perm’: Izd. Perm. Gos. Univ., 1984. P. 110 – 120 [in Russian].

25. Orlov A. I. Artificial intelligence: statistical methods of data analysis: textbook. — Moscow: IPR Media, 2022. — 843 p. [in Russian].

26. Orlov A. I. Statistics of interval data (summarizing article) / Zavod. Lab. Diagn. Mater. 2015. Vol. 81. N 3. P. 61 – 69 [in Russian].

27. Orlov A. I., Lutsenko E. V. System fuzzy interval mathematics. Monograph (scientific publication). — Krasnodar, KubGAU, 2014. — 600 p. [in Russian].

28. Lumel’skii Ya. P. On the issue of comparison of unbiased and other estimates / Applied statistics: Collection of works. — Moscow: Nauka, 1983. P. 316 – 319 [in Russian].

29. Butov A. A., Volkov M. A., Makarov V. P., Orlov A. I., Sharov V. D. Automated system for forecasting and preventing aviation accidents during the organization and production of air transportation / Izv. Samar. NTs RAN. 2012. Vol. 14. N 4(2). P. 380 – 385 [in Russian].