The weighted characteristic function of the multivariate PIT: Independence and goodness-of-fit tests

Many authors have exploited the fact that the distribution of the multivariate probability integral transformation (PIT) of a continuous random vector $X\in R^d$ with cumulative distribution function $F_X$ is free of the marginal distributions. While most of these methods are based on the cdf of $W=F_X\left(X\right)$, this paper introduces the weighted characteristic function (WCf) of $W$. A sample version of the WCf of $W$ based on pseudo-observations is proposed and its weak limit in a space of complex functions is formally established. This result can be used to define test statistics for multivariate independence and goodness-of-fit in copula models, whose asymptotic behaviour comes from the weak convergence of the empirical WCf process. Simulations show the good sampling properties of these new tests, and an illustration is given on the multivariate Cook and Johnson dataset.