Copula-Based Regression Estimation and Inference

Abstract

In this paper we investigate a new approach of estimating a regression function based on copulas.The main idea behind this approach is to write the regression function in terms of a copula and marginal distributions. Once the copula and the marginal distributions are estimated we use the plug-in method to construct the new estimator. Because various methods are available in the literature for estimating both a copula and a distribution, this idea provides a rich and flexible alternative to many existing regression estimators. We provide some asymptotic results related to this copula-based regression modeling when the copula is estimated via profile likelihood and the marginals are estimated nonparametrically. We also study the finite sample performance of the estimator and illustrate its usefulness by analyzing data from air pollution studies. * H. Noh acknowledges financial support from IAP research network P6/03 of the Belgian Government (Belgian Science Policy). † A. El Ghouch acknowledges financial support from IAP research network P6/03 of the Belgian Government (Belgian Science Policy), and from the contract 'Projet d'Actions de Recherche Concertées' (ARC) 11/16-039 of the 'Communauté française de Belgique', granted by the 'Académie universitaire Louvain'.

Some Comments on Copula-Based Regression

“…A common feature of all publications in this direction consists in additional structural or parametric assumptions regarding the unknown regression function, such as additivity (see Stone 1985), tree-based models (Hastie, Tibshirani, and Friedman 2001) or single index models (Ichimura 1993). In a recent article, Noh, El Ghouch, and Bouezmarni (2013) introduced a novel semiparametric estimate of the regression function in a nonparametric regression model with a high-dimensional predictor. Roughly speaking, these authors proposed to model the dependency structure between the response and the predictor by a parametric copula family to obtain estimates of the regression function which converge with a parametric rate.…”

“…, x d ) T and C is the copula. Noh, El Ghouch, and Bouezmarni (2013) showed that the mean regression function m(x 1 , . .…”

“…. , F d , Noh, El Ghouch, and Bouezmarni (2013) suggested to estimate the marginal distributions F Y and F j nonparametrically byF Y (y) = 1 n+1…”

Calibration and Multiple Robustness When Data Are Missing Not At Random

“…The Gaussian copula regression model has been widely used and well studied in the classical low-dimensional setting [40,7,24,30]. For example, [24] developed a systematic framework to make inference and implement model validation for the Gaussian copula regression model.…”

“…For example, [24] developed a systematic framework to make inference and implement model validation for the Gaussian copula regression model. [30] proposed a plug-in approach for estimating a regression function based on copulas, and presented the asymptotic normality of the estimator. However, their model and analysis are restricted to the low-dimensional setting and not well adapted to the high-dimensional case.…”