ANALYSIS OF THE PROPERTIES OF STATISTICAL ESTIMATES OF PARAMETERS OF THE GENERALIZED GAMMA DISTRIBUTION

UDC 519.2

  • Volk Anatoliy Matveevich – PhD (Engineering), Associate Professor, Assistant Professor, the Department of Higher Mathematics. Belarusian State Technological University (13a, Sverdlova str., 220006, Minsk, Republic of Belarus). E-mail: volk@belstu.by

DOI: https://doi.org/10.52065/2520-6141-2023-266-1-2

Keywords: generalized Gamma distribution, physical processes, reliability theory, properties, numerical characteristics, statistical estimation of parameters, best likelihood method, Fisher information matrix, asymptotic efficiency, uniqueness.

For citation: Volk A. M. Analysis of properties of statistical estimates of parameters of the generalized gamma distribution. Proceedings of BSTU, issue 3, Physics and Mathematics. Informatics, 2023, no. 1 (266), pp. 10–14. DOI: https://doi.org/10.52065/2520-6141-2023-266-1-2.

Abstract

A generalized gamma distribution is considered. This distribution generalizes Gamma class distributions and has wide application in statistical methods of investigation of physical processes, in remote sensing, in reliability theory, in description of disperse composition of crushing particles. Its properties have been investigated and numerical characteristics have been found. Equations for statistical estimation of the parameters of this distribution have been obtained by the method of greatest likelihood method. The Fisher information matrix was found for the obtained estimations, its sign-positivity was shown, which proves their consistency, asymptotically-efficiency and uniqueness.

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References

  1. Stacy E. W. A generalization of the gamma distribution. Ann. Math. Statistics, 1962, vol. 33, pp. 1187–1192.
  2. Kudryavtsev A. A. On the representation of the gamma exponential and generalized negative binomial distributions. Informatika i yeye primeneniya [Computer science and its applications], 2019, vol. 13, issue 4, pp. 76–80 (In Russian).
  3. Korolev V. Yu., Krylov V. A., Kuzmin V. Yu. Stability of finite mixtures of generalized gamma distributions with respect to perturbations of parameters. Informatika i yeye primeneniya [Computer science and its applications], 2011, vol. 5, issue 1, pp. 31–38 (In Russian).
  4. Kudryavtsev A. A. A priori generalized gamma distribution in Bayesian balance models. Informatika i yeye primeneniya [Computer science and its applications], 2019, vol. 13, issue 3, pp. 27–33 (In Russian).
  5. Zaks L. M., Korolev V. Yu. Generalized dispersion gamma distributions as the limit for random sums. Informatika i yeye primeneniya [Computer science and its applications], 2013, vol. 7, issue 1, pp. 105–115 (In Russian).
  6. Kouzov P. A. Osnovy analiza dispersionnogo sostava promyshlennykh pyley i izmel'chennykh materialov [Principles of analysis of variance and composition of industrial dust from grinding materials]. Leningrad, Khimiya Publ., 1987. 264 p. (In Russian).
  7. Levdanskiy E. I., Volk A. M., Plekhov I. M. On the particle distribution law in crushing. Tekhnicheskiye osnovy khimicheskoy tekhnologii [Technical Foundations of Chemical Engineering], 1986, no 5, pp. 672–677 (In Russian).
  8. Volk A. M. Generalized Gamma-distribution. Aktual'nyye problemy informatiki: sb. trudov VI Mezhdunoy nauch. konf., 26–30 okt. 1998. V 3 ch. Ch. 2 [Actual problems of informatics: collection of works of the VI International scientific conference 26–30 October 1998. In 3 parts. Part 2]. Minsk, BGU Publ., 1998, pp. 426–432 (In Russian).
  9. Johnson N. L., Kotz S., Balakrishnan N. Odnomernyy'e neprery'vnyy'e raspredeleniya. V 2 ch. Ch. 1 [One-dimensional continuous distributions. In 2 parts. Part 1]. Moscow, BINOM. Laboratoriya znaniy Publ., 2010. 703 p. (In Russian).
  10. Yanke E., Emde F., Lesh F. Spetsial'nyye funktsii: Formuly, grafiki, tablitsy [Special functions: Formulas, graphs, tables]. Moscow, Nauka Publ., 1977. 458 p. (In Russian).
  11. Kramer G. Matematicheskiye metody statistiki: Osnovy modelirovaniya i pervichnaya obrabotka dannykh [Mathematical Methods of Statistics: Basics of modeling and primary data processing]. Moscow, Mir Publ., 1975. 648 p. (In Russian).
  12. Lehman E. Teoriya tochechnogo otsenivaniya [The theory of point estimation]. Moscow, Nauka. Gl. red. fiz.-mat. lit. Publ., 1991. 448 p. (In Russian).
06.02.2023