Geodesic
On the behavior of extreme values in the case of Burr distribution
Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 3 (2024), pp. 18-32
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The paper considers the asymptotic behavior of the moments of extreme values of an organization subjected to risk in the case when the number of factors leading to loss is random. Burr distribution is considered as loss distribution. An asymptotic comparison of the activities of such organizations is carried out in terms of the necessary additional number of such factors (asymptotic deficiency). Two examples illustrating tne obtained results are presented. These examples concern truncated Poisson and binomial distributions.
Keywords: reserve of insurance company, sample of random size, Burr distribution, asymptotic expansions, truncated Poisson and binomial distributions, extreme order statistics, asymptotic deficiency. 
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V. E. Bening. On the behavior of extreme values in the case of Burr distribution. Vestnik Tverskogo gosudarstvennogo universiteta. Seriâ Prikladnaâ matematika, no. 3 (2024), pp. 18-32. http://geodesic.mathdoc.fr/item/VTPMK_2024_3_a1/

[1] Burr I. W., “Cumulative frequency functions”, Annals of Mathematical Statistics, 13 (1942), 215–232 | DOI | MR | Zbl

[2] Bening V. E., “On the asymptotic behavior of the reserve of an organization at risk”, Herald of Tver State University. Series: Applied Mathematics, 2023, no. 4, 25–42 (in Russian) | DOI

[3] Bening V. E., “On the application of the Burr distribution to an asymptotic study of the behavior of an insurance company's reserve”, Bulletin of the Moscow University. Series 15: Computational Mathematics and Cybernetics, 2024, no. 3, 42–56 (in Russian)

[4] Gnedenko B. V., “On the estimation of unknown distribution parameters for a random number of independent observations”, Proceedings of the Tbilisi Mathematical Institute, 92 (1989), 146–150 (in Russian) | Zbl

[5] Gnedenko B. V., Fakhim Kh., “On a transfer theorem”, Doklady Mathematics, 187 (1969), 15–17 (in Russian) | MR | Zbl

[6] Bening V. E., Korolev V. Y., “On an application of the Student distribution in the theory of probability and mathematical statistics”, Theory of Probability and its Applications, 49:3 (2005), 377–391 | DOI | MR | Zbl

[7] Bening V. E., Korolev V. Yu., Generalized Poisson Models and Their Applications in Insurance and Finance, VSP, Utrecht, 2002, 434 pp. | Zbl

[8] Bening V. E., Korolev V. Yu., “Some statistical problems related to the Laplace distribution”, Informatics and Applications, 2:2 (2008), 19–34 (in Russian)

[9] Bening V. E., “Transfer theorems concerning asymptotic expansions for the distribution functions of statistics based on samples with random sizes”, Advanced Studies in Contemporary Mathematics, 28:2 (2018), 187–200 | MR | Zbl

[10] Bening V. E., “On the asymptotic deficiency of some statistical estimators based on samples with random sizes”, Proceedings of the Jangjeon Mathematical Society, 21:2 (2018), 185–193 | MR | Zbl

[11] Bening V. E., “On asymptotic behavior of quantiles of the distributions of statistics based on the samples with random sizes”, Herald of Tver State University. Series: Applied Mathematics, 2017, no. 3, 5–12 (in Russian) | DOI

[12] Bening V. E., “On asymptotic behavior of quantiles deficiencies of the distributions of statistics based on the samples with random sizes”, Herald of Tver State University. Series: Applied Mathematics, 2018, no. 3, 42–57 (in Russian) | DOI

[13] Bening V. E., “On the asymptotic behavior of insurance company reserve”, Herald of Tver State University. Series: Applied Mathematics, 2020, no. 2, 35–48 (in Russian) | DOI

[14] Bening V. E., “On the organizations' risk reserves comparison based on the deficiency concept”, Herald of Tver State University. Series: Applied Mathematics, 2022, no. 3, 5–26 (in Russian)

[15] Bening V., Korolev V., Sukhareva N., Xiaoyang H., Khaydarpashich R., “The Burr distribution as an asymptotic law for extreme order statistics and its application to the analysis of statistical regularities in the interplanetary magnetic field”, Russian Journal of Numerical Analysis and Mathematical Modelling, 39:2 (2024), 61–74 | DOI | MR

[16] Hakim A. R., Fithriani M., Novita M., “Properties of Burr distribution and its application to heavy - tailed survival time data”, Journal of Physics: Conference Series, 1725 (2021), 012016 | DOI

[17] Kendall M., Stuart A., The Advanced Theory of Statistics, Charles Griffin, London, 1945, 457 pp. | MR

[18] Lehmann E. L., Casella G., Theory of Point Estimation, Springer, Berlin, 1998, 589 pp. | MR | Zbl

[19] Hodges J. L., Lehmann E. L., “Deficiency”, The Annals of Mathematical Statistics, 41:5 (1970), 783–801 | DOI | MR | Zbl

[20] Bening V. E., Asymptotic Theory of Testing Statistical Hypotheses: Efficient Statistics, Optimality, Power Loss, and Deficiency, Walter de Gruyter, Berlin, 2011, 277 pp.

[21] Kramer G., Mathematical methods of statistics, Mir Publ., Moscow, 1976, 648 pp. (in Russian)

[22] Znidaric M., Asymptotic expansion for inverse moments of binomial and Poisson distributions, 2005 | DOI | MR