Estimating the spectral measure of a multivariate stable distribution via spherical harmonic analysis

Journal of Multivariate Analysis

Volumn 87, Issue 2, 2003, Pages 219-240

  Pivato, Marcus ^a   Seco, Luis ^b  

^a TRENT UNIVERSITY   (Canada)

^b UNIVERSITY OF TORONTO   (Canada)

Author keywords

Infinitely divisible distributions; Parameter estimation; Spectral measure; Spherical harmonics; Stable probability distributions

Indexed keywords

EID: 0345492357     PISSN: 0047259X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0047-259X(03)00052-6     Document Type: Article

References (61)

1. R. Adler, R. Feldman, M. Taqqu (Eds.), A Practical Guide to Heavy Tailed Data, Bïrkhauser, Boston, 1998.
2. A. Araujo, E. Giné, The Central Limit Theorem for Real and Banach Valued Random Variables, Wiley, New York, 1980.
3. V.S. Bawa, E.J. Elton, M.J. Gruber, Simple rules for optimal portfolio selection in stable paretian markets, J. Finance 34 (1979) 1041-1047.
4. T. Brocker, T. tom Dieck, Representations of Compact Lie Groups, Springer, New York, 1985.
5. S. Cambanis, G. Miller, Linear problems in pth order stable processes, SIAM J. Appl. Math. 43 (1) (1981) 43-69.
6. I. Chavel, Riemannian Geometry: A Modern Introduction, Cambridge University Press, Cambridge, MA, 1993.
7. B.N. Cheng, S.T. Rachev, Multivariate stable future prices, Math. Finance 5 (2) (1995) 133-153.
8. G. Christoph, W. Wolf, Convergence Theorems with a Stable Limit Law, Akadamie, Berlin, 1993.
9. R.R. Coifman, G. Wiess, Representations of compact groups and spherical harmonics, L'Enseignment Math. 14 (1968) 123-173.
10. J.B. Conway, A Course in Functional Analysis, 2nd Edition, Springer, New York, 1990.
11. R. Courant, D. Hilbert, Methods of Mathematical Physics, Vol. I, Wiley-Interscience, New York, 1961.
12. H. Dym, P. McKean, Fourier Series and Integrals, Academic Press, New York, 1972.
13. P. Embrechts, C. Klüppelberg, T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer, Heidelberg, 1997.
14. A. Erdelyi, et al., Higher Transcendental Functions, Vol. I, II, III, McGraw-Hill, New York, 1953-1955.
15. L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.
16. E.F. Fama, Mandelbrot and the stable Paretian hypothesis, J. Business (Chicago) 36 (1963) 420-429.
17. E.F. Fama, The distribution of daily stock prices: a test of Mandelbrot's stable Paretian hypothesis, J. Business (Chicago) 38 (1965) 34-105.
18. E.F. Fama, Portfolio analysis in a stable Paretian market, Management Sci. 6 (1965) 404-419.
19. E. Feldheim, Etude de la stabilité des lois de probabilitié, Ph.D. Thesis, Thése de la Faculté des Sciences de Paris, 1937.
20. W. Feller, An Introduction to Probability Theory and its Applications, 2nd Edition, Vol. II, Wiley, New York, 1966.
21. B.D. Fielitz, E.W. Smith, Asymmetric stable distributions of stock price change, J. Amer. Statist. Assoc. (Appl.) 67 (340) (1972) 813-814.
22. A. Gamba, Portfolio analysis with symmetric stable paretian returns, in: E. Canestrelli (Ed.), Current Topics in Quantitative Finance, Physica, Heidelberg, 1999, pp. 48-69.
23. A. Gamba, Portfolio analysis with symmetric stable paretian returns, in: Current Topics in Quantitative Finance, Contributions in Management Science, Physica, Heidelberg, 1999.
24. I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series, and Products, Corrected and enlarged edition, Academic Press, Toronto, 1980.
25. S. Helgason, Topics in Harmonic Analysis on Homogeneous Spaces, Birkhäuser, Boston, MA, 1981.
26. J. Holtsmark, Über die verbreiterung von spektrallinien, Ann. Physik 4 (58 363) (1919) 577-630.
27. T. Hurd, J. Boland, M. Pivato, L. Seco, Measures of dependence for multivariate lévy distributions, in: Disordered and Complex Systems, Vol. 553 of American Institute of Physics Conference Proceedings (London), 2000, pp. 289-295.
28. A. Janicki, A. Weron, Simulation and Chaotic Behaviour of α-stable Stochastic Processes, Marcel Dekker, New York, 1994.
29. J. Kuelbs, A representation theorem for symmetric stable processes and symmetric measures on h, Z. Wahrscheinlichtkeitstheorie verwalte Gebiete 26 (1973) 259-271.
30. N.N. Lebedev, Special Functions and Applications, Dover, New York, 1972.
31. W.E. Leland, M.S. Taqqu, W. Willinger, On the self-similar nature of ethernet traffic, IEEE/ACM Trans. Network. 2 (1994) 1-15.
32. B.B. Mandelbrot, The variation of certain speculative prices, in: Fractals and Scaling in Finance: Discontinuities, Concentration, Risk, Springer, New York, 1965.
33. R.N. Mantegna, H.E. Stanley, Scaling behaviour in the dynamics of an economic index, Nature 376 (6) (1995) 46-49.
34. J.H. McCulloch, Financial applications of stable distributions, in: Statistical Methods in Finance, Vol. 14 of Handbook of Statistics, North-Holland, Amsterdam, 1996, pp. 393-425.
35. J.P. Nolan, A.K. Panorska, J.H. McCulloch, Estimation of stable spectral measures, Stable non-Gaussian models in finance and econometrics, Math. Comput. Modelling 34 (9-11) (2001) 1113-1122.
36. C. Müller, Spherical Harmonics, in: Lecture Notes in Mathematics, Vol. 17, Springer, New York, 1966.
37. C.L. Nikias, A.P. Petropulu, Higher Order Spectral Analysis: A Nonlinear Signal Processing Environment, Prentice-Hall, Englewood Cliffs, NJ, 1993.
38. C.L. Nikias, M. Shao, Signal Processing with Alpha-Stable Distributions and Applications, Wiley, New York, 1995.
39. J.P. Nolan, Multivariate stable distributions: approximation, estimation, simulation, and identification, 1999 (preprint), available from http://www.cas.american.edu/~jpnolan.
40. J.P. Nolan, Stable distributions, April 1999 (preprint), available from http://www.cas.american.edu/~jpnolan.
41. J.P. Nolan, Anna K. Panorska, Data analysis for heavy tailed multivariate samples, Comm. Statist. Stochastic Models 13 (4) (1997) 687-702.
42. R.D. Peirce, Application of the positive α-stable distribution, in: Signal Processing Workshop on Higher-Order Statistics, IEEE Press, Piscataway, New Jersey, 1997, pp. 420-424.
43. M. Pivato, Analytical Methods for Multivariate Stable Probability Distributions, Ph.D. Thesis, University of Toronto, 2001, available at http://xaravve.trentu.ca/pivato/Research/thesis.pdf.
44. J.S. Press, Estimation in univariate and multivariate stable distributions, J. Amer. Statist. Assoc. (Theory and Methods) 67 (340) (1972) 842-846.
45. J.S. Press, Multivariate stable distributions, J. Multivariate Anal. 2 (1972) 444-462.
46. S.T. Rachev, Huang Xin, Test for association of random variables in the domain of attraction of multivariate stable law, Probab. Math. Statist. 14 (1) (1993) 125-141.
47. G. Samorodnitsky, M.S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman & Hall, New York, 1994.
48. P.A. Samuelson, Efficient portfolio selection for pareto-levy investments, J. Financial Quantitat. Anal. 2 (1967) 107-122.
49. B.W. Stuck, B. Kleiner, A statistical analysis of telephone noise, Bell Systems Tech. J. 53 (1974) 1263-1320.
50. M. Sugiura, Unitary Representations and Harmonic Analysis, Wiley, New York, 1975.
51. M. Takeuchi, Modern Spherical Functions, Vol. 135 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1994.
52. J.D. Talman, Special Functions: A Group Theoretic Approach, Benjamin, New York, 1968.
53. M.E. Taylor, Noncommutative Harmonic Analysis, American Mathematical Society, Providence, RI, 1986.
54. A. Terras, Harmonic Analysis on Symmetric Spaces and Applications, Vol. I, Springer, New York, 1985.
55. J.S. Bodenschatz, C.L. Nikias, Maximum-likelihood symmetric α-stable parameter estimation, IEEE Trans. Signal Process. 47 (5) (1999) 1382-1384.
56. N.J. Vilenkin, Special Functions and the Theory of Group Representations, Vol. 22 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1968.
57. F.M. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer, New York, 1983.
58. W. Willinger, V. Paxson, Where mathematics meets the internet, Notices Amer. Math. Soc. 45 (8) (1998) 961-970.
59. W. Willinger, M.S. Taqqu, A. Erramilli, A bibliographical guide to self-similar traffic and performance modelling for modern high-speed networks, in: F.P. Kelly, S. Zachary, I. Ziedins (Eds.), Stochastic Networks: Theory and Applications, Clarendon Press, Oxford, UK, 1996, pp. 339-366.
60. W. Willinger, M. Taqqu, R. Sherman, D. Wilson, Self-similarity through high variability: statistical analysis of ethernet LAN traffic at the source level, in: Proceedings of the fourth IEEE International Symposium on High Performance Distributed Computing, Washington, DC, 1995, pp. 100-113.
61. V.M. Zolotarev, One-dimensional Stable Distribution, American Mathematical Society, Providence, RI, 1986.