Testing the equality of the laws of two strictly stationary processes

In this paper we consider the problem of comparison of two strictly stationary processes. The novelty of our approach is that we consider all their d-dimensional joint distributions, for d⩾ 1. Our procedure consists in expanding their densities in a multivariate orthogonal basis and comparing their k first coefficients. The dimension d to consider and the number k of coefficients to compare in view of performing the test can growth with the sample size and are automatically selected by a two-step data-driven procedure. The method works for possibly paired, short or long range dependent processes. A simulation study shows the good behavior of the test procedure. In particular, we apply our method to compare ARFIMA processes. Some real-life applications also illustrate this approach.