Goodness-of-Fit tests with Dependent Observations

Abstract

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.

26 pages

References in corpus (1)

• Principal Regression Analysis and the index leverage effect

Cited by in corpus (8)

• Dynamical Hurst exponent as a tool to monitor unstable periods in financial time series
• Statistical laws in linguistics
• Testing statistical laws in complex systems
• The joint distribution of stock returns is not elliptical
• A Kolmogorov-Smirnov test for the molecular clock on Bayesian ensembles of phylogenies
• Copulas and time series with long-ranged dependences
• On Hawkes Processes with Infinite Mean Intensity
• Radical Complexity