Goodness-of-Fit tests with Dependent Observations
AbstractWe 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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Bibliographic InfoPaper provided by arXiv.org in its series Papers with number 1106.3016.
Date of creation: Jun 2011
Date of revision: Aug 2011
Publication status: Published in J. Stat. Mech. (2011) P09003
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Web page: http://arxiv.org/
This paper has been announced in the following NEP Reports:
- NEP-ALL-2011-06-25 (All new papers)
- NEP-ECM-2011-06-25 (Econometrics)
- NEP-ETS-2011-06-25 (Econometric Time Series)
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- R\'emy Chicheportiche & Jean-Philippe Bouchaud, 2013. "Some applications of first-passage ideas to finance," Papers 1306.3110, arXiv.org.
- 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.
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