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Parameter estimation of selfsimilarity exponents

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  • Becker-Kern, Peter
  • Pap, Gyula
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    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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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 99 (2008)
    Issue (Month): 1 (January)
    Pages: 117-140

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    Handle: RePEc:eee:jmvana:v:99:y:2008:i:1:p:117-140

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    Keywords: Operator semi-selfsimilar process Ornstein-Uhlenbeck type process Parameter estimation Selfsimilarity exponent Spectral decomposition;

    References

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    1. Wolfe, Stephen James, 1983. "Continuity properties of decomposable probability measures on euclidean spaces," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 13(4), pages 534-538, December.
    2. Yamazato, Makoto, 1983. "Absolute continuity of operator-self-decomposable distributions on Rd," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 13(4), pages 550-560, December.
    3. Meerschaert, Mark M. & Scheffler, Hans-Peter, 1999. "Moment Estimator for Random Vectors with Heavy Tails," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 71(1), pages 145-159, October.
    4. Maejima, Makoto & Sato, Ken-iti & Watanabe, Toshiro, 2000. "Distributions of selfsimilar and semi-selfsimilar processes with independent increments," Statistics & Probability Letters, Elsevier, Elsevier, vol. 47(4), pages 395-401, May.
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