Quantile spectral processes

Abstract

Quantile- and copula-related spectral concepts recently have been considered by various authors. Those spectra, in their most general form, provide a full characterization of the copulas associated with the pairs (Xt;Xt-k) in a process (Xt)t2Z, and account for important dynamic features, such as changes in the conditional shape (skewness, kurtosis), time-irreversibility, or dependence in the extremes, that their traditional counterpart cannot capture. Despite various proposals for estimation strategies, no asymptotic distributional results are available so far for the proposed estimators, which constitutes an important obstacle for their practical application. In this paper, we provide a detailed asymptotic analysis of a class of smoothed rank-based cross-periodograms associated with the copula spectral density kernels introduced in Dette et al. (2011). We show that, for a very general class of (possibly non-linear) processes, properly scaled and centered smoothed versions of those crossperiodograms, indexed by couples of quantile levels, converge weakly, as stochastic processes, to Gaussian processes. A first application of those results is the construction of asymptotic confi dence intervals for copula spectral density kernels. The same convergence results also provide asymptotic distributions (under serially dependent observations) for a new class of rank-based spectral methods involving the Fourier transforms of rank-based serial statistics such as the Spearman, Blomqvist or Gini autocovariance coefficients.