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

References

[1] A. Balkema and L. de Haan, Residual life time at great age. Annals of Probability 2 (1974), 792–804.10.1214/aop/1176996548Search in Google Scholar

[2] K. Burnecki, W. Härdle, and R. Weron, An Introduction to Simulation of Risk Processes. Hugo Steinhaus Center, Wroclaw University of Technology, Wroclaw, 2004. Available at http://prac.im.pwr.edu.pl/hugo/RePEc/wuu/wpaper/HSC_03_04.pdf10.1002/9780470012505.tas013mSearch in Google Scholar

[3] I. W. Burr, Cumulative frequency functions. Annals of Mathematical Statistics 13 (1942), No. 2, 215–232.10.1214/aoms/1177731607Search in Google Scholar

[4] I. W. Burr, The effect of non-normality on constants for X and R charts. Industrial Quality Control (1967), 563–569.Search in Google Scholar

[5] I. W. Burr and P. J. Cislak, On a general system of distributions, II. The sample median. Journal of the American Statistical Association 63 (1968), 627–635.10.1080/01621459.1968.11009281Search in Google Scholar

[6] I. W. Burr, On a general system of distributions, III. The sample range. Journal of the American Statistical Association 63 (1968), 636–643.10.1080/01621459.1968.11009282Search in Google Scholar

[7] I. W. Burr, Parameters for a general system of distributions to match a grid of α₃ and α₄. Communications in Statistics 2 (1973), 1–21.10.1080/03610917308548217Search in Google Scholar

[8] D. R. Cook and E. S. Johnson, Generalized Burr–Pareto-logistic distributions with applications to an uranium exploration data set. Technometrics 28 (1986), No. 2, 123–131.10.1080/00401706.1986.10488113Search in Google Scholar

[9] M. Ghitany and S. Al-Awadhi, Maximum likelihood estimation of Burr XII distribution parameters under random censoring. Journal of Applied Statistics 29 (2002), No. 7, 955–965.10.1080/0266476022000006667Search in Google Scholar

[10] L. J. Gleser, The gamma distribution as a mixture of exponential distributions. American Statistician 43 (1989), 115–117.10.1080/00031305.1989.10475632Search in Google Scholar

[11] B. D. Gnedenko and V. V. Korolev, Random Summation: Limit Theorems and Applications. Boca Raton, CRC Press, 1996.Search in Google Scholar

[12] J. Grandell, Doubly Stochastic Poisson Processes. Lecture Notes Math. 529. Springer, Berlin–Heidelberg, 1976.10.1007/BFb0077758Search in Google Scholar

[13] J. Grandell, Mixed Poisson Processes. Chapman and Hall, London, 1997.10.1007/978-1-4899-3117-7Search in Google Scholar

[14] M. E. Grigor’eva, V. Yu. Korolev, and I. A. Sokolov, A limit theorem for geometric sums of independent nonidentically distributed random variables and its application to the prediction of the probabilities of catastrophes in nonhomogeneous flows of extremal events. Informatics and Its Applications 7 (2013), No. 4, 11–19.Search in Google Scholar

[15] P. L. Gupta, R. C. Gupta, and S. Lvin, Analysis of failure time data by Burr distribution. Communications in Statistics, Theory and Methods 25 (1976), 2013–2024.10.1080/03610929608831817Search in Google Scholar

[16] A. R. Hakim, I. Fithriani, and M. Novita, Properties of Burr distribution and its application to heavytailed survival time data. Journal of Physics: Conference Series 1725 (2021), Art. 012016.10.1088/1742-6596/1725/1/012016Search in Google Scholar

[17] M. A. Hatke, A certain cumulative probability function. Annals of Mathematical Statistics 20 (1949), 461–463.10.1214/aoms/1177730002Search in Google Scholar

[18] V. Yu. Korolev, Probability and Statistical Methods for the Decomposition of the Volatility of Chaotic Processes. Moscow University Publishing House, Moscow, 2011.Search in Google Scholar

[19] V. Yu. Korolev, Analogs of Gleser’s theorem for negative binomial and generalized gamma distributions and some their applications. Informatics and Its Applications 11 (2017), 2–17.Search in Google Scholar

[20] V. Yu. Korolev, I. A. Sokolov, and A. K. Gorshenin, Max-compound Cox processes, I. Journal of Mathematical Sciences 237 (2019), 789–803.10.1007/s10958-019-04205-0Search in Google Scholar

[21] V. Korolev, Some properties of univariate and multivariate exponential power distributions and related topics. Mathematics 8 (2020), No. 11, Art. 1918.10.3390/math8111918Search in Google Scholar

[22] V. Yu. Korolev, I. A. Sokolov, and A. K. Gorshenin, Max-compound Cox processes, II. Journal of Mathematical Sciences 246 (2020), 488–502.10.1007/s10958-020-04754-9Search in Google Scholar

[23] V. Yu. Korolev, I. A. Sokolov, and A. K. Gorshenin, Max-compound Cox processes, III. Journal of Mathematical Sciences 267 (2022), 273–288.10.1007/s10958-022-06133-ySearch in Google Scholar

[24] V. Korolev and A. Zeifman, Mixture representations for generalized Burr, Snedecor–Fisher and generalized Student distributions with related results. Mathematics 11 (2023), Art. 398210.3390/math11183892Search in Google Scholar

[25] V. Korolev, Analytic and asymptotic properties of the generalized Student and generalized Lomax distributions. Mathematics 11 (2023), Art. 2890, p. 1–27.10.3390/math11132890Search in Google Scholar

[26] W. C. Lee, J. W. Wu, and C. W. Hong, Assessing the lifetime performance index of products from progressively type II right censored data using Burr XII model. Mathematics and Computers In Simulation 79 (2009), 2167–2179.10.1016/j.matcom.2008.12.001Search in Google Scholar

[27] A. W. Lewis, The Burr distribution as a general parametric family in survivorship and reliability theory applications. PhD Dissertation. Chapel Hill: Department of Biostatistics, University of North Carolina at Chapel Hill, 1981.Search in Google Scholar

[28] S. R. Lindsay, G. R. Wood, and R. C. Woollons, Modelling the diameter distribution of forest stands using the Burr distribution. Journal of Applied Statistics 23 (1996), No. 6, 609–619.10.1080/02664769623973Search in Google Scholar

[29] J. B. McDonald, Some generalized functions of the size distribution of income. Econometrica 52 (1984), No. 3, 647–663.10.2307/1913469Search in Google Scholar

[30] J. B. McDonald and D. O. Richards, Model selection, some generalized distributions. Communications in Statistics, Theory and Methods 16 (1987), 1049–1047.10.1080/03610928708829422Search in Google Scholar

[31] P. W. Mielke Jr. and E. S. Johnson, Some generalized beta distributions of the second kind having desirable applications features in hydrology and meterology. Water Resources Research 10 (1974), 223–226.10.1029/WR010i002p00223Search in Google Scholar

[32] K. J. Parker and S. S. Poul, Generalized formulations producing a Burr distribution of speckle statistics. Journal of Medical Imaging 9 (2022), No. 2, Art. 023501.10.1117/1.JMI.9.2.023501Search in Google Scholar PubMed PubMed Central

[33] J. Pickands, Statistical inference using extreme order statistics. Annals of Statistics 3 (1975), 119–131.10.1214/aos/1176343003Search in Google Scholar

[34] N. A. Rodriguez, Guide to the Burr type XII distribution. Biometrika 6 (1977), No. 4, 129–134.10.1093/biomet/64.1.129Search in Google Scholar

[35] R. N. Rodriguez and B. Y. Taniguchi, A new statistical model for predicting customer octane satisfaction using trained-rater observations. SAE Technical Paper 801356. Transactions of the Society of Automotive Engineers (1980), 4213–4240.10.4271/801356Search in Google Scholar

[36] Q. X. Shao, H. Wong, J. Xia, and W. C. Ip, Models for extremes using the extended three-parameter Burr XII system with application to flood frequency analysis. Hydrological Sciences Journal 49 (2004), No. 4, 685–702.10.1623/hysj.49.4.685.54425Search in Google Scholar

[37] S. Singh and G. Maddala, A function for the size distribution of incomes. Econometrica 44 (1976), No. 5, 963–970.10.2307/1911538Search in Google Scholar

[38] P. R. Tadikamalla, A look at the Burr and related distributions. International Statistical Review. Revue Internationale de Statistique 48 (1980), No. 3, 337–344.10.2307/1402945Search in Google Scholar

[39] J. L. Teugels, A note on Poisson-subordination. The Annals of Mathematical Statistics 43 (1972), 676–680.10.1214/aoms/1177692653Search in Google Scholar

[40] D. R. Wingo, Maximum likelihood estimation of the Burr XII distribution parameters under type-II censoring. Microelectronic Reliability 23 (1993), 1251–1257.10.1016/0026-2714(93)90126-JSearch in Google Scholar

[41] W. J. Zimmer, J. B. Keats, and F. K. Wang, The Burr XII distribution in reliability analysis. Journal of Quality Technology 30 (1998), 386–394.10.1080/00224065.1998.11979874Search in Google Scholar