Estimating the scale parameter of a Lévy-stable distribution via the extreme value approach. (English) Zbl 1130.62051
Summary: The characteristic exponent \(\alpha\) of a Lévy-stable law \(S_{\alpha}(\sigma, \beta, \mu)\) was thoroughly studied as the extreme value index of a heavy tailed distribution. For \(1 < \alpha < 2\), L. Peng [Stat. Probab. Lett. 52, No. 3, 255–264 (2001; Zbl 0981.62041)] has proposed, via the extreme value approach, an asymptotically normal estimator for the location parameter \(\mu\). We derive by the same approach, an estimator for the scale parameter \(\sigma\) and we discuss its limiting behavior.
MSC:
| 62G32 | Statistics of extreme values; tail inference |
| 62G05 | Nonparametric estimation |
| 62G20 | Asymptotic properties of nonparametric inference |
| 60E07 | Infinitely divisible distributions; stable distributions |
| 65C60 | Computational problems in statistics (MSC2010) |
Citations:
Zbl 0981.62041Software:
RReferences:
| [1] | J. M. Chambers, C. L. Mallows, and B. W. Stuck, ”A method for simulating stable random variables,” Journal of the American Statistical Association vol. 71 pp. 340–344, 1976. · Zbl 0341.65003 · doi:10.2307/2285309 |
| [2] | S. Cheng and L. Peng, ”Confidence intervals for the tail index,” Bernoulli vol. 7 pp. 751–760, 2001. · Zbl 0985.62039 · doi:10.2307/3318540 |
| [3] | J. Danielsson, L. de Haan, L. Peng, and C. G. de Vries, ”Using a bootstrap method to choose the sample fraction in tail index estimation,” Journal of Multivariate Analysis vol. 76 pp. 226–248, 2001. · Zbl 0976.62044 · doi:10.1006/jmva.2000.1903 |
| [4] | A. L. M. Dekkers and L. de Haan, Optimal choice of sample fraction in extreme-value estimation, Journal of Multivariate Analysis vol. 47 pp. 173–195, 1993. · Zbl 0797.62016 · doi:10.1006/jmva.1993.1078 |
| [5] | H. Drees, ”Optimal rates of convergence for estimates of the extreme value index,” Annals of Statistics vol. 26 pp. 434–448, 1998. · Zbl 0934.62047 · doi:10.1214/aos/1030563992 |
| [6] | W. H. DuMouchel, ”On the asymptotic normality of the maximum-likelihood estimate when sampling from a stable distribution,” Annals of Statistics vol. 1 no. 5 pp. 948–957, 1973. · Zbl 0287.62013 · doi:10.1214/aos/1176342516 |
| [7] | E. F. Fama, ”The behavior of stock market prices,” Journal of Business vol. 38 pp. 34–105, 1965. · Zbl 0129.11903 · doi:10.1086/294743 |
| [8] | A. Fereira and C. G. de Vries, Optimal confidence intervals for the tail index and high quantiles. Tinberg Institute Discussion Paper 090/2, 2004. |
| [9] | M. I. Fraga Alves, ”Estimation of the tail parameter in the domain of attraction of an extremal distribution,” Journal of Statistical Planning and Inference vol. 45, pp. 143–173, 1995. · Zbl 0820.62032 · doi:10.1016/0378-3758(94)00068-9 |
| [10] | J. Geluk and L. de Haan, Regular variation, extensions and Tauberian theorems. In Mathematical Centre Tracts, vol. 40, Centre for Mathematics and Computer Science, Amsterdam, 1987. · Zbl 0624.26003 |
| [11] | L. de Haan and U. Stadtmüller, ”Generalized regular variation of second order,” Journal of the Australian Mathematical Society, Series A vol. 61 pp. 381–395, 1996. · Zbl 0878.26002 · doi:10.1017/S144678870000046X |
| [12] | P. Hall, ”Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems,” Journal of Multivariate Analysis vol. 32 pp. 177–203, 1990. · Zbl 0722.62030 · doi:10.1016/0047-259X(90)90080-2 |
| [13] | P. Hall and A. H. Welsh, ”Best attainable rates of convergence for estimates of parameters of regular variation,” Annals of Statistics vol. 12 pp. 1079–1084, 1984. · Zbl 0539.62048 · doi:10.1214/aos/1176346723 |
| [14] | P. Hall and A. H. Welsh, ”Adaptive estimates of parameters of regular variation,” Annals of Statistics vol. 13 331–341, 1985. · Zbl 0605.62033 · doi:10.1214/aos/1176346596 |
| [15] | W. Härdle, S. Klinke, and M. Müller, XploRe Learning Guide, Springer, Berlin Heidelberg New York, 2000. |
| [16] | B. Hill, ”A simple approach to inference about tail of a distribution,” Annals of Statistics vol. 3 pp. 1163–1174, 1975. · Zbl 0323.62033 · doi:10.1214/aos/1176343247 |
| [17] | R. Ihaka, R. Gentleman, ”R: a language for data analysis and graphics,” Journal of Computational and Graphical Statistics vol. 5 pp. 299–314, 1996. · doi:10.2307/1390807 |
| [18] | S. M. Kogon and D. B. Williams, ”Characteristic function based estimation of stable parameters.” In R. Adler, R. Feldman, M. Taqqu (eds.), A Practical Guide to Heavy Tails, pp. 311–335, Birkhauser, Cambridge, MA, 1998. · Zbl 0946.62019 |
| [19] | I. A. Koutrouvelis, ”Regression-type estimation of the parameters of stable law,” Journal of the American Statistical Association vol. 75 pp. 918–928, 1980. · Zbl 0449.62026 · doi:10.2307/2287182 |
| [20] | R. G., Laha and V. K. Rohatgi, Probability Theory, Wiley, New York, 1979. |
| [21] | P. Lévy, Calcul des Probabilités, Gauthier-Villars, Paris, 1925. |
| [22] | B. Mandelbrot, ”The variation of certain speculative prices,” Journal of Business vol. 36 pp. 394–419, 1963. · doi:10.1086/294632 |
| [23] | J. H. McCulloch, ”Simple consistent estimators of stable distribution parameters,” Communications in Statistics. Simulation and Computation vol. 15 pp. 1109–1136, 1986. · Zbl 0612.62028 · doi:10.1080/03610918608812563 |
| [24] | A. Necir, ”A nonparametric sequential test with power 1 for the mean of Lévy-stable distributions with infinite variance,” Methodology and Computing in Applied Probability vol. 8 pp. 321–343, 2006. · Zbl 1103.62074 · doi:10.1007/s11009-006-9749-9 |
| [25] | C. Neves and M. I. Fraga Alves, ”Reiss and Thomas automatic selection of the number of extremes,” Computational Statistics and Data Analysis vol. 47 pp. 689–704, 2004. · Zbl 1430.62096 · doi:10.1016/j.csda.2003.11.011 |
| [26] | J. P. Nolan, ”Maximum likelihood estimation and diagnostics for stable distributions.” In O. E. Barndorff-Nielsen, T. Mikosch, S. Resnick (eds.), Lévy Processes, Brikhäuser, Boston, MA, 2001. · Zbl 0971.62008 |
| [27] | D. Ojeda, Comparison of stable estimators. Ph.D. Thesis, Department of Mathematics and Statistics, American University, 2001. |
| [28] | L. Peng, ”Asymptotically unbiased estimators for the extreme-value index,” Statistics & Probability Letters vol. 38 pp. 107–115, 1998. · Zbl 1246.62129 · doi:10.1016/S0167-7152(97)00160-0 |
| [29] | L. Peng, ”Estimating the mean of a heavy tailed distribution,” Statistics & Probability Letters vol. 52 pp. 255–264, 2001. · Zbl 0981.62041 · doi:10.1016/S0167-7152(00)00203-0 |
| [30] | S. Rachev, Handbook of Heavy-tailed Distributions in Finance, North Holland, Amsterdam, 2003. |
| [31] | R. D. Reiss and M. Thomas, Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields. Birkhäuser, Basel, 1997. · Zbl 0880.62002 |
| [32] | G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, New York, 1994. · Zbl 0925.60027 |
| [33] | S. Stoyanov and B. Racheva-Iotova, ”Univariate stable laws in the field of finance – parameter estimation,” Journal of Concrete and Applicable Mathematics vol. 2 no. 4 pp 24–49, 2004. · Zbl 1110.62148 |
| [34] | X. Wei, ”Asymptotically efficient estimation of the index of regular variation,” Annals of Statistics vol. 23 pp. 2036–2058, 1995. · Zbl 0854.62022 · doi:10.1214/aos/1034713646 |
| [35] | R. Weron, ”Levy-stable distributions revisited: tail index > 2 does not exclude the Levy-stable regime,” International Journal of Modern Physics, C vol. 12 no. 2 pp. 209–223, 2001. · doi:10.1142/S0129183101001614 |
| [36] | R. Weron, ”Computationally intensive Value at Risk calculations.” In J. E. Gentle, W. Härdle, Y. Mori (eds.) Handbook of Computational Statistics, pp. 911–950, Springer, Berlin Heidelberg New York, 2004. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
