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Goodness-of-Fit tests with Dependent Observations

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  • Remy Chicheportiche
  • Jean-Philippe Bouchaud
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    Abstract

    We revisit the Kolmogorov-Smirnov and Cram\'er-von Mises goodness-of-fit (GoF) tests and propose a generalisation to identically distributed, but dependent univariate random variables. We show that the dependence leads to a reduction of the "effective" number of independent observations. The generalised GoF tests are not distribution-free but rather depend on all the lagged bivariate copulas. These objects, that we call "self-copulas", encode all the non-linear temporal dependences. We introduce a specific, log-normal model for these self-copulas, for which a number of analytical results are derived. An application to financial time series is provided. As is well known, the dependence is to be long-ranged in this case, a finding that we confirm using self-copulas. As a consequence, the acceptance rates for GoF tests are substantially higher than if the returns were iid random variables.

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    File URL: http://arxiv.org/pdf/1106.3016
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1106.3016.

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    Date of creation: Jun 2011
    Date of revision: Aug 2011
    Publication status: Published in J. Stat. Mech. (2011) P09003
    Handle: RePEc:arx:papers:1106.3016

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    Web page: http://arxiv.org/

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    Cited by:
    1. Morales, Raffaello & Di Matteo, T. & Gramatica, Ruggero & Aste, Tomaso, 2012. "Dynamical generalized Hurst exponent as a tool to monitor unstable periods in financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(11), pages 3180-3189.
    2. R\'emy Chicheportiche & Jean-Philippe Bouchaud, 2013. "Some applications of first-passage ideas to finance," Papers 1306.3110, arXiv.org.

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