## Bernoulli

• Bernoulli
• Volume 22, Number 3 (2016), 1770-1807.

### Quantile spectral processes: Asymptotic analysis and inference

#### 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 $(X_{t},X_{t-k})$ in a process $(X_{t})_{t\in\mathbb{Z}}$, 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 counterparts cannot capture. Despite various proposals for estimation strategies, only quite incomplete 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. [Bernoulli 21 (2015) 781–831]. We show that, for a very general class of (possibly nonlinear) processes, properly scaled and centered smoothed versions of those cross-periodograms, 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 confidence 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.

#### Article information

Source
Bernoulli, Volume 22, Number 3 (2016), 1770-1807.

Dates
Revised: February 2015
First available in Project Euclid: 16 March 2016

https://projecteuclid.org/euclid.bj/1458132999

Digital Object Identifier
doi:10.3150/15-BEJ711

Mathematical Reviews number (MathSciNet)
MR3474833

Zentralblatt MATH identifier
1369.62245

#### Citation

Kley, Tobias; Volgushev, Stanislav; Dette, Holger; Hallin, Marc. Quantile spectral processes: Asymptotic analysis and inference. Bernoulli 22 (2016), no. 3, 1770--1807. doi:10.3150/15-BEJ711. https://projecteuclid.org/euclid.bj/1458132999

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#### Supplemental materials

• Supplement to “Quantile spectral processes: Asymptotic analysis and inference”. We provide details for the proof of part (ii) of Theorem 3.6, and proofs for Propositions 3.1, 3.2, and 3.4. Further, we prove results from Section A.4, namely Lemmas A.1–A.7.