Importance of Generalized Logistic Distribution in Extreme Value Modeling

Affiliation(s)

Indian Institute of Science, Bangalore, India.

University of Calicut, Calicut, India.

Indian Institute of Science, Bangalore, India.

University of Calicut, Calicut, India.

ABSTRACT

We consider a problem from stock market modeling, precisely, choice of adequate distribution of modeling extremal behavior of stock market data. Generalized extreme value (GEV) distribution and generalized Pareto (GP) distribution are the classical distributions for this problem. However, from 2004, [1] and many other researchers have been empirically showing that generalized logistic (GL) distribution is a better model than GEV and GP distributions in modeling extreme movement of stock market data. In this paper, we show that these results are not accidental. We prove the theoretical importance of GL distribution in extreme value modeling. For proving this, we introduce a general multivariate limit theorem and deduce some important multivariate theorems in probability as special cases. By using the theorem, we derive a limit theorem in extreme value theory, where GL distribution plays central role instead of GEV distribution. The proof of this result is parallel to the proof of classical extremal types theorem, in the sense that, it possess important characteristic in classical extreme value theory, for e.g. distributional property, stability, convergence and multivariate extension etc.

Cite this paper

K. Nidhin and C. Chandran, "Importance of Generalized Logistic Distribution in Extreme Value Modeling,"*Applied Mathematics*, Vol. 4 No. 3, 2013, pp. 560-573. doi: 10.4236/am.2013.43080.

K. Nidhin and C. Chandran, "Importance of Generalized Logistic Distribution in Extreme Value Modeling,"

References

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[24] K. Tolikas, and G. D. Gettinby, “Modelling the Distribution of the Extreme Share Returns in Singapore,” Journal of Empirical Finance, Vol. 16, No. 2, 2009, pp. 254-263. doi:10.1016/j.jempfin.2008.06.006

[25] K. Tolikas, “Value-at-Risk and Extreme Value Distributions for Financial Returns,” The Journal of Risk, Vol. 10, No. 3, 2008, pp. 31-77.

[26] S. I. Resnick, “Tail Equivalence and Its Applications,” Journal of Applied Probability, Vol. 8, 1971, pp. 135-156. doi:10.2307/3211844

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[32] K. Nidhin and C. Chandran, “Limit Theorems of General Functions of Independent and Identically Distributed Random Variables,” Statistics and Probability Letters, Vol. 79, No. 23, 2009, pp. 2397-2404. doi:10.1016/j.spl.2009.08.013

[33] L. B. Klebanov, S. Mitinik, S. T. Rachev and V. E. Volkovic, “A New Representation for the Characteristic Function of Strictly Geo-stable Vectors,” Journal of Applied Probability, Vol. 37, No. 4, 1999, pp. 1137-1142.

[34] T. T. Nguyen and A. R. Sampson, “A Note on Characterizations of Multivariate Stable Distributions,” Annals of the Institute of Statistical Mathematics, Vol. 43, No. 4, 1991, pp. 793-801. doi:10.1007/BF00121655

[35] J. Galambos, “The Asymptotic Theory of Extreme Order Statistics,” Wiley, New York, 1978.

[36] M. A. Stephens, “EDF Statistics for Goodness of Fit and Some Comparisons,” Journal of American Statistical Association, Vol. 69, No. 347, 1974, pp. 730-737. doi:10.1080/01621459.1974.10480196

[37] K. Nidhin and C. Chandran, “An Analysis of the Extremal Behavior of Bombay Stock Exchange Data,” International Journal of Statistics and Analysis, Vol. 1, No. 3, 2011, pp. 239-256.

[38] T. J. Kozubowski and S. T. Rachev, “Multivariate Geometric Stable Laws,” Journal of Computational Analysis and Applications, Vol. 1, No. 4, 1999, pp. 349-385. doi:10.1023/A:1022692806500

[39] J. L. Kelley, “General Topology,” Springer, New York, 1955.

[40] S. Lipschutz, “General Topology,” Schaum’s Outline Series, New York, 1965. doi:10.1007/978-1-4612-5449-2

[1] G. D. Gettinby, C. D. Sinclair, D. M. Power and R. A. Brown, “An Analysis of the Distribution of Extremes Share Returns in the UK from 1975 to 2000,” Journal of Business Finance and Accounting, Vol. 31, No. 5-6, 2004, pp. 607-645. doi:10.1111/j.0306-686X.2004.00551.x

[2] P. Kearns and A. Pagan, “Estimating the Density Tail Index for Financial Time Series,” Review of Economic Statistics, Vol. 79, No. 2, 1997, pp. 171-175. doi:10.1162/003465397556755

[3] J. Danielson, P. Hartmann and C. de Vries, “The Cost of Conservatism,” Risk, Vol. 11, No. 1, 1998, pp. 103-107.

[4] J. Cotter, “Margin Exceedences for European Stock Index futures Using Extreme Value Theory,” Journal of Banking & Finance, Vol. 25, No. 8, 2001, pp. 1475-1502. doi:10.1016/S0378-4266(00)00137-0

[5] E. Fama, “The Behaviour of Stock Market Price,” Journal of Business, Vol. 38, No. 1, 1965, pp. 34-105. doi:10.1086/294743

[6] J. B. Gray and D. W. French, “Emperical Comparisons of Distributional Models for Stock Index Returns,” Journal of Business, Finance and Accounting, Vol. 17, No. 3, 1990, pp. 451-459. doi:10.1111/j.1468-5957.1990.tb01197.x

[7] R. D. F. Harris and C. C. Kucukozmen, “The Emperical Distribution of UK and US Stock Returns,” Journal of Business Finance and Accounting, Vol. 28, No. 5-6, 2001, pp. 715-740. doi:10.1111/1468-5957.00391

[8] B. Mandelbrot, “The Variation of Certain Speculative Prices,” Journal of Business, Vol. 36, No. 4, 1963, pp. 394-419. doi:10.1086/294632

[9] J. McDonald and Y. Xu, “A generalization of Beta Distribution with Applications,” Journal of Econometrics, Vol. 66, No. 1-2, 1995, pp. 133-152. doi:10.1016/0304-4076(94)01612-4

[10] A. Peiro, “The Distribution of Stock Returns: Internatinal Evidence,” Applied Financial Economics, Vol. 4, No. 6, 1994, pp. 431-439. doi:10.1080/758518675

[11] P. Theodossiou, “Financial Data and the Skewed Generalised-T Distribution,” Management Science, Vol. 44, No. 12, 1998, pp. 1650-1661. doi:10.1287/mnsc.44.12.1650

[12] P. Embrechts, C. Kluppelberg and T. Mikosch, “Modeling Extremal Events for Insurance and Finance,” Springer-Verlang, Berling, Heidelberg, 1997. doi:10.1007/978-3-642-33483-2

[13] S. Kotz and S. Nadarajah, “Extreme Value Distributions: Theory and Applications,” Imperial Collage Press, London, 1999.

[14] S. I. Resnick, “Extreme Values, Regular Variation, and Point Processes,” Springer-Verlag, New York, 1987.

[15] A. A. Balkama and L. de Haan, “Residual Life Time at Great Age,” The Annals of Probability, Vol. 2, No. 5, 1974, pp. 792-804. doi:10.1214/aop/1176996548

[16] J. Pickands, “Statistical Inference Using Extreme Order Statistics,” The Annals of Statistics, Vol. 3, No. 1, 1975, pp. 119-131. doi:10.1214/aos/1176343003

[17] H. Rootzen and N. Taijvidi, “Multivarate Generalized Pareto Distribution,” Bernoulli, Vol. 12, No. 5, 2006, pp. 917-930. doi:10.3150/bj/1161614952

[18] M. R. Leadbetter, G. Lindgren and H. Rootzen, “Extremes and Related Properties of Random Sequences and Processes,” Springer-Verlag, New York, 1983.

[19] F. M. Longin, “The Asymptotic Distribution of Extreme Stock Market Returns,” Journal of Business, Vol. 69, No. 3, 1996, pp. 383-408. doi:10.1086/209695

[20] F. M. Longin, “From Value at Risk to Stress Testing: The Extreme Value Approach,” Journal of Banking and Finance, Vol. 24, No. 7, 2000, pp. 1097-1130. doi:10.1016/S0378-4266(99)00077-1

[21] F. M. Longin, “Stock Market Crashes: Some Quantitative Results Based on Extreme Value Theory, Derivatives Use,” Trading Regulation, Vol. 7, No. 3, 2001, pp. 197-205.

[22] A. I. Maghyereh and H. A. Al-Zoubi, “The Tail Behavior of Extreme Stock Returns in the Gulf Emerging Markets: An Implication for Financial Risk Management,” Studies in Economics and Finance, Vol. 25, No. 1, 2008, pp. 21-37. doi:10.1108/10867370810857540

[23] K. Tolikas and R. A. Brown, “The Distribution of Extreme Daily Share Returns in the Athens Stock Exchange,” The European Journal of Finance, Vol. 12, No. 1, 2006, pp. 1-12. doi:10.1080/1351847042000304107

[24] K. Tolikas, and G. D. Gettinby, “Modelling the Distribution of the Extreme Share Returns in Singapore,” Journal of Empirical Finance, Vol. 16, No. 2, 2009, pp. 254-263. doi:10.1016/j.jempfin.2008.06.006

[25] K. Tolikas, “Value-at-Risk and Extreme Value Distributions for Financial Returns,” The Journal of Risk, Vol. 10, No. 3, 2008, pp. 31-77.

[26] S. I. Resnick, “Tail Equivalence and Its Applications,” Journal of Applied Probability, Vol. 8, 1971, pp. 135-156. doi:10.2307/3211844

[27] W. J. Voorn, “Characterization of the Logistic and Loglogistic Distributions by Extreme Value Related Stability with Random Sample Size,” Journal of Applied Probability, Vol. 24, 1987, pp. 838-851. doi:10.2307/3214209

[28] N. Balakrishnan, “Handbook of the Logistic Distribution,” Marcel Dekker, New York, 1992.

[29] N. L. Johnson, S. Kotz and N. Balakrishnan, “Continuous Univariate Distributions,” John Wiley, New York, 1995.

[30] J. R. M. Hosking and J. R. Wallis, “Regional Frequency Analysis: An Approach Based on L-Moments,” Cambridge University Press, Cambridge, 1997.

[31] S. Hongjoon, N. Woosung, J. Younghun and H. JunHaeng, “Asymptotic Variance of Regional Curve for Generalized Logistic Distribution,” World Environmental and Water Resources Congress, Great Rivers, 2009.

[32] K. Nidhin and C. Chandran, “Limit Theorems of General Functions of Independent and Identically Distributed Random Variables,” Statistics and Probability Letters, Vol. 79, No. 23, 2009, pp. 2397-2404. doi:10.1016/j.spl.2009.08.013

[33] L. B. Klebanov, S. Mitinik, S. T. Rachev and V. E. Volkovic, “A New Representation for the Characteristic Function of Strictly Geo-stable Vectors,” Journal of Applied Probability, Vol. 37, No. 4, 1999, pp. 1137-1142.

[34] T. T. Nguyen and A. R. Sampson, “A Note on Characterizations of Multivariate Stable Distributions,” Annals of the Institute of Statistical Mathematics, Vol. 43, No. 4, 1991, pp. 793-801. doi:10.1007/BF00121655

[35] J. Galambos, “The Asymptotic Theory of Extreme Order Statistics,” Wiley, New York, 1978.

[36] M. A. Stephens, “EDF Statistics for Goodness of Fit and Some Comparisons,” Journal of American Statistical Association, Vol. 69, No. 347, 1974, pp. 730-737. doi:10.1080/01621459.1974.10480196

[37] K. Nidhin and C. Chandran, “An Analysis of the Extremal Behavior of Bombay Stock Exchange Data,” International Journal of Statistics and Analysis, Vol. 1, No. 3, 2011, pp. 239-256.

[38] T. J. Kozubowski and S. T. Rachev, “Multivariate Geometric Stable Laws,” Journal of Computational Analysis and Applications, Vol. 1, No. 4, 1999, pp. 349-385. doi:10.1023/A:1022692806500

[39] J. L. Kelley, “General Topology,” Springer, New York, 1955.

[40] S. Lipschutz, “General Topology,” Schaum’s Outline Series, New York, 1965. doi:10.1007/978-1-4612-5449-2