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Parameter estimation of selfsimilarity exponents

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  • Becker-Kern, Peter
  • Pap, Gyula

Abstract

The characteristic feature of operator selfsimilar stochastic processes is that a linear rescaling in time is equal in the sense of distributions to a linear operator rescaling in space, which in turn is characterized by the selfsimilarity exponent. The growth behaviour of such processes in any radial direction is determined by the real parts of the eigenvalues of the selfsimilarity exponent. We extend an estimation method of Meerschaert and Scheffler [M.M. Meerschaert, H.-P. Scheffler, Moment estimator for random vectors with heavy tails, J. Multivariate Anal. 71 (1999) 145-159, M.M. Meerschaert, H.-P. Scheffler, Portfolio modeling with heavy tailed random vectors, in: S.T. Rachev (Ed.), Handbook of Heavy Tailed Distributions in Finance, Elsevier Science B.V., Amsterdam, 2003, pp. 595-640] to be applicable for estimating the real parts of the eigenvalues of the selfsimilarity exponent and corresponding spectral directions given by the eigenvectors. More generally, the results are applied to operator semi-selfsimilar processes, which obey a weaker scaling property, and to certain Ornstein-Uhlenbeck type processes connected to operator semi-selfsimilar processes via Lamperti's transformation.

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  • Becker-Kern, Peter & Pap, Gyula, 2008. "Parameter estimation of selfsimilarity exponents," Journal of Multivariate Analysis, Elsevier, vol. 99(1), pages 117-140, January.
    • Handle: RePEc:eee:jmvana:v:99:y:2008:i:1:p:117-140
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    References listed on IDEAS

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    1. Becker-Kern, Peter, 2004. "Random integral representation of operator-semi-self-similar processes with independent increments," Stochastic Processes and their Applications, Elsevier, vol. 109(2), pages 327-344, February.
    2. Maejima, Makoto & Sato, Ken-iti & Watanabe, Toshiro, 2000. "Distributions of selfsimilar and semi-selfsimilar processes with independent increments," Statistics & Probability Letters, Elsevier, vol. 47(4), pages 395-401, May.
    3. Yamazato, Makoto, 1983. "Absolute continuity of operator-self-decomposable distributions on Rd," Journal of Multivariate Analysis, Elsevier, vol. 13(4), pages 550-560, December.
    4. Jeanblanc, M. & Pitman, J. & Yor, M., 0. "Self-similar processes with independent increments associated with Lévy and Bessel processes," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 223-231, July.
    5. Meerschaert, Mark M. & Scheffler, Hans-Peter, 1999. "Moment Estimator for Random Vectors with Heavy Tails," Journal of Multivariate Analysis, Elsevier, vol. 71(1), pages 145-159, October.
    6. Wolfe, Stephen James, 1983. "Continuity properties of decomposable probability measures on euclidean spaces," Journal of Multivariate Analysis, Elsevier, vol. 13(4), pages 534-538, December.
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    Cited by:

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    3. Gustavo Didier & Vladas Pipiras, 2012. "Exponents, Symmetry Groups and Classification of Operator Fractional Brownian Motions," Journal of Theoretical Probability, Springer, vol. 25(2), pages 353-395, June.
    4. Patrice Abry & Gustavo Didier & Hui Li, 2019. "Two-step wavelet-based estimation for Gaussian mixed fractional processes," Statistical Inference for Stochastic Processes, Springer, vol. 22(2), pages 157-185, July.

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