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, X t−k ) in a process (Xt) t∈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 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 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.
In this paper, we present an alternative method for the spectral analysis of a univariate, strictly stationary time series {Yt} t∈Z . We define a "new" spectrum as the Fourier transform of the differences between copulas of the pairs (Yt, Y t−k ) and the independence copula. This object is called a copula spectral density kernel and allows to separate the marginal and serial aspects of a time series. We show that this spectrum is closely related to the concept of quantile regression. Like quantile regression, which provides much more information about conditional distributions than classical location-scale regression models, copula spectral density kernels are more informative than traditional spectral densities obtained from classical autocovariances. In particular, copula spectral density kernels, in their population versions, provide (asymptotically provide, in their sample versions) a complete description of the copulas of all pairs (Yt, Y t−k ). Moreover, they inherit the robustness properties of classical quantile regression, and do not require any distributional assumptions such as the existence of finite moments. In order to estimate the copula spectral density kernel, we introduce rank-based Laplace periodograms which are calculated as bilinear forms of weighted L1-projections of the ranks of the observed time series onto a harmonic regression model. We establish the asymptotic distribution of those periodograms, and the consistency of adequately smoothed versions. The finite-sample properties of the new methodology, and its potential for applications are briefly investigated by simulations and a short empirical example.
In this paper a new nonparametric estimate of conditional quantiles is proposed, that
The empirical copula process plays a central role in the asymptotic analysis of many statistical procedures which are based on copulas or ranks. Among other applications, results regarding its weak convergence can be used to develop asymptotic theory for estimators of dependence measures or copula densities, they allow to derive tests for stochastic independence or specific copula structures, or they may serve as a fundamental tool for the analysis of multivariate rank statistics. In the present paper, we establish weak convergence of the empirical copula process (for observations that are allowed to be serially dependent) with respect to weighted supremum distances. The usefulness of our results is illustrated by applications to general bivariate rank statistics and to estimation procedures for the Pickands dependence function arising in multivariate extreme-value theory.
A collection of quantile curves provides a complete picture of conditional distributions. A properly centered and scaled version of estimated curves at various quantile levels gives rise to the so-called quantile regression process (QRP). In this paper, we establish weak convergence of QRP in a general series approximation framework, which includes linear models with increasing dimension, nonparametric models and partial linear models. An interesting consequence is obtained in the last class of models, where parametric and non-parametric estimators are shown to be asymptotically independent. Applications of our general process convergence results include the construction of non-crossing quantile curves and the estimation of conditional distribution functions. As a result of independent interest, we obtain a series of Bahadur representations with exponential bounds for tail probabilities of all remainder terms. Bounds of this kind are potentially useful in analyzing statistical inference procedures under divide-and-conquer setup.MSC 2010 subject classifications: Primary 62F12, 62G20, 62G08.4. New bounds for local basis functions: Last but not the least, in Sections 2.2 and 5.2, we provide results for models with "local basis structure" (for instance B-splines). For such basis functions, we show that the conditions on model dimension can be relaxed from m 4 = o(n 1−ε ) [required by Theorem 12 of Belloni et al. (2016)] to m 2 (log n) 6 = o(n) in the case of B-splines [see the discussion below Assumption (B1)].Given the discussions above, we would also like to point out that Belloni et al. (2016) discuss other aspects such as bootstrap approximations which are not covered in our paper. In summary, both Belloni et al. (2016) and the present paper consider the same model setup, but focus on different aspects of the resulting theory, and none of the two papers is more general than the other.The rest of this paper is organized as follows. Section 2 presents the weak convergence of QRP under general series approximation framework. Section 3 discusses the QRP in quantile partial linear models. As an application of our weak convergence theory, Section 4 considers various functionals of the quantile regression process. A detailed discussion on our novel Bahadur representations is given in Section 5, and all proofs are deferred to the appendix.
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