Quantile spectral processes: Asymptotic analysis and inference

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.Comment: Published at http://dx.doi.org/10.3150/15-BEJ711 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Empirical likelihood estimation of the spatial quantile regression

The spatial quantile regression model is a useful and flexible model for analysis of empirical problems with spatial dimension. This paper introduces an alternative estimator for this model. The properties of the proposed estimator are discussed in a comparative perspective with regard to the other available estimators. Simulation evidence on the small sample properties of the proposed estimator is provided. The proposed estimator is feasible and preferable when the model contains multiple spatial weighting matrices. Furthermore, a version of the proposed estimator based on the exponentially tilted empirical likelihood could be beneficial if model misspecification is suspect

Efficient prediction for linear and nonlinear autoregressive models

Conditional expectations given past observations in stationary time series are usually estimated directly by kernel estimators, or by plugging in kernel estimators for transition densities. We show that, for linear and nonlinear autoregressive models driven by independent innovations, appropriate smoothed and weighted von Mises statistics of residuals estimate conditional expectations at better parametric rates and are asymptotically efficient. The proof is based on a uniform stochastic expansion for smoothed and weighted von Mises processes of residuals. We consider, in particular, estimation of conditional distribution functions and of conditional quantile functions.Comment: Published at http://dx.doi.org/10.1214/009053606000000812 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Uniform Bahadur Representation for Nonparametric Censored Quantile Regression: A Redistribution-of-Mass Approach

Censored quantile regressions have received a great deal of attention in the literature. In a linear setup, recent research has found that an estimator based on the idea of “redistribution-of-mass” in Efron (1967, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. 4, pp. 831–853, University of California Press) has better numerical performance than other available methods. In this paper, this idea is combined with the local polynomial kernel smoothing for nonparametric quantile regression of censored data. We derive the uniform Bahadur representation for the estimator and, more importantly, give theoretical justification for its improved efficiency over existing estimation methods. We include an example to illustrate the usefulness of such a uniform representation in the context of sufficient dimension reduction in regression analysis. Finally, simulations are used to investigate the finite sample performance of the new estimator

Inference on counterfactual distributions

We develop inference procedures for policy analysis based on regression methods. We consider policy interventions that correspond to either changes in the distribution of covariates, or changes in the conditional distribution of the outcome given covariates, or both. Under either of these policy scenarios, we derive functional central limit theorems for regression-based estimators of the status quo and counterfactual marginal distributions. This result allows us to construct simultaneous confidence sets for function-valued policy effects, including the effects on the marginal distribution function, quantile function, and other related functionals. We use these confidence sets to test functional hypotheses such as no-effect, positive effect, or stochastic dominance. Our theory applies to general policy interventions and covers the main regression methods including classical, quantile, duration, and distribution regressions. We illustrate the results with an empirical application on wage decompositions using data for the United States. Of independent interest is the use of distribution regression as a tool for modeling the entire conditional distribution, encompassing duration/transformation regression, and representing an alternative to quantile regression. This is a revision of CWP09/09.

About adaptive coding on countable alphabets

This paper sheds light on universal coding with respect to classes of memoryless sources over a countable alphabet defined by an envelope function with finite and non-decreasing hazard rate. We prove that the auto-censuring AC code introduced by Bontemps (2011) is adaptive with respect to the collection of such classes. The analysis builds on the tight characterization of universal redundancy rate in terms of metric entropy % of small source classes by Opper and Haussler (1997) and on a careful analysis of the performance of the AC-coding algorithm. The latter relies on non-asymptotic bounds for maxima of samples from discrete distributions with finite and non-decreasing hazard rate

Inference on counterfactual distributions

In this paper we develop procedures for performing inference in regression models about how potential policy interventions affect the entire marginal distribution of an outcome of interest. These policy interventions consist of either changes in the distribution of covariates related to the outcome holding the conditional distribution of the outcome given covariates fixed, or changes in the conditional distribution of the outcome given covariates holding the marginal distribution of the covariates fixed. Under either of these assumptions, we obtain uniformly consistent estimates and functional central limit theorems for the counterfactual and status quo marginal distributions of the outcome as well as other function-valued effects of the policy, including, for example, the effects of the policy on the marginal distribution function, quantile function, and other related functionals. We construct simultaneous confidence sets for these functions; these sets take into account the sampling variation in the estimation of the relationship between the outcome and covariates. Our procedures rely on, and our theory covers, all main regression approaches for modeling and estimating conditional distributions, focusing especially on classical, quantile, duration, and distribution regressions. Our procedures are general and accommodate both simple unitary changes in the values of a given covariate as well as changes in the distribution of the covariates or the conditional distribution of the outcome given covariates of general form. We apply the procedures to examine the effects of labor market institutions on the U.S. wage distribution.

Quantile spectral processes

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