Goodness-of-Fit Tests with Dependent Observations

We revisit the Kolmogorov-Smirnov and Cramér-von Mises goodness-of-fit (GoF) tests and propose a generalization to identically distributed, but dependent uni-variate random variables. We show that the dependence leads to a reduction of the "effective" number of independent observations. The generalized GoF tests are not distribution-free but rather depend on all the lagged bi-variate 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