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Product Representations for Random Variables with Weibull Distributions and Their Applications*

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In this paper, product representations are obtained for random variables with theWeibull distribution in terms of random variables with normal, exponential and stable distributions yielding scale mixture representations for the corresponding distributions. Main attention is paid to the case where the shape parameter γ of theWeibull distribution belongs to the interval (0, 1]. The case of small values of γ is of special interest, since the Weibull distributions with such parameters occupy an intermediate position between distributions with exponentially decreasing tails (e.g., exponential and gamma-distributions) and heavy-tailed distributions with Zipf–Pareto power-type decrease of tails. As a by-product result of the representation of the Weibull distribution with γ ∈ (0, 1) in the form of a mixed exponential distribution, the explicit representation of the moments of symmetric or one-sided strictly stable distributions are obtained. It is demonstrated that if γ ∈ (0, 1], then the Weibull distribution is a mixed half-normal law, and hence, it can be limiting for maximal random sums of independent random variables with finite variances. It is also demonstrated that the symmetric two-sided Weibull distribution with γ ∈ (0, 1] is a scale mixture of normal laws. Necessary and sufficient conditions are proved for the convergence of the distributions of extremal random sums of independent random variables with finite variances and of the distributions of the absolute values of these random sums to the Weibull distribution as well as of those of random sums themselves to the symmetric two-sided Weibull distribution. These results can serve as theoretical grounds for the application of the Weibull distribution as an asymptotic approximation for statistical regularities observed in the scheme of stopped random walks used, say, to describe the evolution of stock prices and financial indexes. Also, necessary and sufficient conditions are proved for the convergence of the distributions of more general regular statistics constructed from samples with random sizes to the symmetric two-sided Weibull distribution.

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References

  1. R. B. Abernethy, The New Weibull Handbook. Reliability and Statistical Analysis for Predicting Life, Safety, Survivability, Risk, Cost and Warranty Claims (5th Edition), Robert B. Abernethy, North Palm Beach (2004).

  2. R. DAddario, “Intorno ad una funzione di distribuzione,” Giorn. Econ. Anna. Econ., 33, 205–214 (1974).

    Google Scholar 

  3. S. N. Antonov and S. N. Koksharov, “On the asymptotic behavior of the tails of scale mixtures of normal distributions,” Statist. Methods Estim. Test. Hyp., 18, 90–105 (2006).

    Google Scholar 

  4. E. B. Bagirov, “Some remarks on mixtures of normal distributions,” Theor. Probab. Appl., 33, No. 4, 778–780 (1988).

    MathSciNet  MATH  Google Scholar 

  5. C. P. A. Bartels, Economic Aspects of Regional Welfare, Martinus Nijhoff, Leiden (1977).

  6. V. Bening and V. Korolev, Generalized Poisson Models and their Applications in Insurance and Finance, VSP, Utrecht (2002).

  7. J. G. Bennett, “Broken coal,” J. Inst. Fuel, 10, 22–39 (1936).

  8. L. Bondesson, “A general result on infinite divisibility,” Ann. Probab., 7, No. 6, 965–979 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  9. R. F. Bordley, J. B. McDonald, and A. Mantrala. “Something new, something old: Parametric models for the size distribution of income,” J. Income Distrib., 6, 91–103 (1996).

    Google Scholar 

  10. G. Box and G. Tiao, Bayesian Inference in Statistical Analysis, Addison–Wesley, New York (1973).

    MATH  Google Scholar 

  11. R. Elandt-Johnson and N. Johnson, Survival Models and Data Analysis, Wiley, New York (1999).

    Book  MATH  Google Scholar 

  12. R. A. Fisher and L. H. C. Tippett, “Limiting forms of the frequency distribution of the largest or smallest member of a sample,” Proc. Cambridge Phil. Soc., 24, 180–190 (1928).

    Article  MATH  Google Scholar 

  13. M. Fréchet, “Sur la loi de probabilité de l’écart maximum,” Ann. Soc. Pol. Math. (Cracovie), 6, 93–116 (1927).

  14. J. Galambos, The Asymptotic Theory of Extreme Order Statistics (2nd Edition), Wiley, New York (1978).

    MATH  Google Scholar 

  15. B. V. Gnedenko, “Sur la distribution limite du terme maximum d’une serie aléatoire,” Ann. Math., 44, No. 3, 423–453 (1943).

    Article  MathSciNet  MATH  Google Scholar 

  16. B. V. Gnedenko, “On estimation of unknown parameters from a random number of independent observations,” Trans. Razmadze Tbilisi Math. Inst., 92, 146–150 (1989).

    MathSciNet  MATH  Google Scholar 

  17. B. V. Gnedenko and V. Yu. Korolev, Random Summation: Limit Theorems and Applications, CRC Press, Boca Raton (1996).

  18. C. M. Goldie, “A class of infinitely divisible distributions,” Math. Proc. Cambridge Philos. Soc., 63, 1141–1143 (1967).

    Article  MathSciNet  MATH  Google Scholar 

  19. I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series and Products (5th Edition), Academic Press, New York (1994).

    Google Scholar 

  20. J. Grandell, Doubly Stochastic Poisson Processes, Springer, Berlin (1976).

    Book  MATH  Google Scholar 

  21. M. E. Grigoryeva and V. Yu. Korolev, “On convergence of the distributions of random sums to skew exponential-power laws,” Inform. Appl., 7, No. 3, 66–74 (2013).

    Google Scholar 

  22. M. E. Grigoryeva, V. Yu. Korolev, and I. A. Sokolov, “A limit theorem for geometric sums of independent non-identically distributed random variables and its application to prediction of probabilities of catastrophes in non-homogeneous flows of extremal events,” Inform. Appl., 7, No. 4, 11–19 (2013).

    Google Scholar 

  23. M. E. Grigoryeva and V. Yu. Korolev, “On some properties of the Weibull distribution and its applicability to the analysis of risks of catastrophically accumulating unfavorable effects in non-homogeneous flows of events,” Stat. Methods Estim. Test. Hyp., 25, 100–119 (2013).

    Google Scholar 

  24. E. Gumbel, Statistics of Extremes, Columbia University Press, New York (1958).

    MATH  Google Scholar 

  25. R. V. Hogg and S. A. Klugman, Loss Distributions, Wiley, New York (1984).

  26. N. L. Johnson and S. Kotz, Continuous Univariate Distributions, Houghton Mifflin Company, Boston (1970).

    MATH  Google Scholar 

  27. N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, 2nd Edition, Wiley, New York (1994).

    MATH  Google Scholar 

  28. V. Yu. Korolev, “Convergence of random sequences with independent random indices. I,” Theor. Probab. Appl., 39, No. 2, 313–333 (1994).

    MathSciNet  MATH  Google Scholar 

  29. V. Yu. Korolev, “Convergence of random sequences with independent random indices. II,” Theor. Probab. Appl., 40, No. 4, 907–910 (1995).

    MathSciNet  MATH  Google Scholar 

  30. V. Yu. Korolev, “A general theorem on the limit behavior of superpositions of independent random processes with applications to Cox processes,” J. Math. Sci., 81, No. 5, 2951–2956 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  31. V. Yu. Korolev, “Asymptotic properties of extrema of compound Cox processes and their applications to some problems of financial mathematics,” Theor. Probab. Appl., 45, No. 1, 182–194 (2000).

    MathSciNet  MATH  Google Scholar 

  32. V. Yu. Korolev and I. A. Sokolov, Mathematical Models of Non-Homogeneous Flows of Extremal Events, Torus Press, Moscow (2008).

    Google Scholar 

  33. V. Yu. Korolev, Probability and Statistical Methods for Decomposition of the Volatility of Chaotic Processes, Moscow University Publishing House, Moscow (2011).

    MATH  Google Scholar 

  34. V. Yu. Korolev, V. E. Bening, and S. Ya. Shorgin, Mathematical Foundations of Risk Theory (2nd Edition), FIZMATLIT, Moscow (2011).

    MATH  Google Scholar 

  35. V. Yu. Korolev, M. E. Grigoryeva, Yu. S. Nefedova, and R. Lazovskii, “A method of estimation of probabilities of catastrophes in non-homogeneous flows of extremal events and its application to prediction of water floods in Saint-Petersburg,” Actuary, 5, No. 1, 53–58 (2014).

  36. S. Kotz and S. Nadarajah, Extreme value distributions. Theory and Applications, Imperial College Press, London (2000).

  37. J. Laherrére and D. Sornette, “Stretched exponential distributions in nature and economy: ’fat tails’ with characteristic scales,” Eur. Phys. J., B, 2, 525–539 (1998).

    Article  Google Scholar 

  38. J. F. Lawless, Statistical Models and Methods for Lifetime Data, Wiley, New York (1982).

    MATH  Google Scholar 

  39. Y. Malevergne, V. Pisarenko, and D. Sornette, “Empirical distributions of stock returns: Between the stretched exponential and the power law?,” Quant. Fin., 5, 379–401 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  40. Y. Malevergne, V. Pisarenko, and D. Sornette, “On the power of generalized extreme value (GEV) and generalized Pareto distribution (GPD) estimators for empirical distributions of stock returns,” Appl. Fin. Econ., 16, 271–289 (2006).

    Article  Google Scholar 

  41. S. Mittnik and S. T. Rachev, “Stable distributions for asset returns,” Appl. Math. Let., 2, 301–304 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  42. S. Mittnik and S. T. Rachev, “Modeling asset returns with alternative stable distributions,” Econ. Rev., 12, 261–330 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  43. Qian Chen and R. H. Gerlach, The Two-Sided Weibull Distribution and Forecasting Financial Tail Risk, OME Working Paper No. 01/2011, Business School, The University of Sydney, Sydney (2011).

  44. Qian Chen and R. H. Gerlach, “The two-sided Weibull distribution and forecasting financial tail risk,” Int. J. Forecast., 29, No. 4, 527–540 (2013).

  45. J. W. S. Rayleigh, “On the resultant of a large number of vibrations of the same pitch and of arbitrary phase,” Philos. Mag., 5th Ser., 10, 73–78 (1880).

  46. P. Rosin and E. Rammler, “The laws governing the fineness of powdered coal,” J. Inst. Fuel, 7, 29–36 (1933).

    Google Scholar 

  47. P. Rosin, E. Rammler, and K. Sperling, Korngróßenprobleme des Kohlenstaubes und ihre Bedeutung für die Vermahlung. Bericht C 52 des Reichskohlenrates, VDI-Verlag, Berlin (1933).

  48. M. Nawaz Sharif and M. Nazrul Islam, “The Weibull distribution as a general model for forecasting technological change,” Tech. Forecast. Soc. Change, 18, No. 3, 247–256 (1980).

    Article  Google Scholar 

  49. D. Sornette, P. Simonetti, and J. V. Andersen, “Φq-field theory for portfolio optimization: fat-tails and non-linear correlations,” Phys. Report., 335, No. 2, 19–92 (2000).

    Article  Google Scholar 

  50. D. Stoyan, “Weibull, RRSB or extreme-value theorists?,” Metrika, 76, 153–159 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  51. W. Weibull, A Statistical Theory Of The Strength Of Materials, Ingeniörsvetenskapsakademien-Handlingar, Nr. 151, Generalstabens Litografiska Anstalts Förlag, Stockholm (1939).

  52. W. Weibull, The Phenomenon of Rupture in Solids, Ingeniörsvetenskapsakademien-Handlingar, Nr. 153, Generalstabens Litografiska Anstalts Förlag, Stockholm (1939).

  53. W. Weibull, “A statistical distribution function of wide applicability,” ASME J. Appl. Mech., 18, No. 3, 293–297 (1951).

    MATH  Google Scholar 

  54. V. M. Zolotarev, One-Dimensional Stable Distributions, American Mathematical Society, Providence (1986).

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Authors and Affiliations

  1. Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia

    V. Yu. Korolev

  2. Federal Research Center “Informatics and Control,” Russian Academy of Sciences, Moscow, Russia

    V. Yu. Korolev

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  1. V. Yu. Korolev
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Correspondence to V. Yu. Korolev.

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* Research supported by the Russian Foundation for Basic Research, project 15–07–04040.

Proceedings of the XXXII International Seminar on Stability Problems for Stochastic Models, Trondheim, Norway, June 16–21, 2014

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Korolev, V.Y. Product Representations for Random Variables with Weibull Distributions and Their Applications*. J Math Sci 218, 298–313 (2016). https://doi.org/10.1007/s10958-016-3031-7

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