2011/06/30 by Remy Chicheportiche, Jean-Philippe Bouchaud · 1 citation
Economics, Econometrics and Finance · Physics and Astronomy · Mathematics · #q-fin.ST #cond-mat.stat-mech #stat.AP #msc:91B84 #msc:62P20 #msc:62M10 #msc:60F05
paper · pdf · doi:10.1088/1742-5468/2011/09/p09003
published as J. Stat. Mech. (2011) P09003 · 26 pages
arxiv created 2011/08/03 · arxiv updated 2011/09/06
We revisit the Kolmogorov-Smirnov and Cramér-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.