Anouar El Ghouch

Copula-Based Regression Estimation and Inference

We investigate a new approach to 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 our 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 family of 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.

Local Polynomial Quantile Regression With Parametric Features

We propose a new approach to conditional quantile function estimation that combines both parametric and nonparametric techniques. At each design point, a global, possibly incorrect, pilot parametric model is locally adjusted through a kernel smoothing fit. The resulting quantile regression estimator behaves like a parametric estimator when the latter is correct and converges to the nonparametric solution as the parametric start deviates from the true underlying model. We give a Bahadur-type representation of the proposed estimator from which consistency and asymptotic normality are derived under an α -mixing assumption. We also propose a practical bandwidth selector based on the plug-in principle and discuss the numerical implementation of the new estimator. Finally, we investigate the performance of the proposed method via simulations and illustrate the methodology with a data example.

Semiparametric copula quantile regression for complete or censored data

When facing multivariate covariates, general semiparametric regression techniques come at hand to propose flexible models that are unexposed to the curse of dimensionality. In this work a semiparametric copula-based estimator for conditional quantiles is investigated for complete or right-censored data. In spirit, the methodology is extending the recent work of Noh et al. (2013) and Noh et al. (2015), as the main idea consists in appropriately defining the quantile regression in terms of a multivariate copula and marginal distributions. Prior estimation of the latter and simple plug-in lead to an easily implementable estimator expressed, for both contexts with or without censoring, as a weighted quantile of the observed response variable. In addition, and contrary to the initial suggestion in the literature, a semiparametric estimation scheme for the multivariate copula density is studied, motivated by the possible shortcomings of a purely parametric approach and driven by the regression context. The resulting quantile regression estimator has the valuable property of being automatically monotonic across quantile levels, and asymptotic normality for both complete and censored data is obtained under classical regularity conditions. Finally, numerical examples as well as a real data application are used to illustrate the validity and finite sample performance of the proposed procedure.

EMPIRICAL LIKELIHOOD CONFIDENCE INTERVALS FOR DEPENDENT DURATION DATA

Three types of confidence intervals are developed for a general class of functionals of a survival distribution based on censored dependent data. The confidence intervals are constructed via asymptotic normality (Wald’s method), the empirical likelihood (EL) method, and the blockwise EL method in which sample means over blocks of observations are used in place of the original data. Asymptotic results are derived to accurately calibrate the various procedures, and their performance is evaluated in a simulation study. The problem of the choice of the block size is also discussed.

Semiparametric Conditional Quantile Estimation through Copula-Based Multivariate Models

We consider a new approach in quantile regression modeling based on the copula function that defines the dependence structure between the variables of interest. The key idea of this approach is to rewrite the characterization of a regression quantile in terms of a copula and marginal distributions. After the copula and the marginal distributions are estimated, the new estimator is obtained as the weighted quantile of the response variable in the model. The proposed conditional estimator has three main advantages: it applies to both iid and time series data, it is automatically monotonic across quantiles, and, unlike other copula-based methods, it can be directly applied to the multiple covariates case without introducing any extra complications. We show the asymptotic properties of our estimator when the copula is estimated by maximizing the pseudo-log-likelihood and the margins are estimated nonparametrically including the case where the copula family is misspecified. We also present the finite sample performance of the estimator and illustrate the usefulness of our proposal by an application to the historical volatilities of Google and Yahoo.

Bernstein estimator for unbounded copula densities

Abstract Copulas are widely used for modeling the dependence structure of multivariate data. Many methods for estimating the copula density functions are investigated. In this paper, we study the asymptotic properties of the Bernstein estimator for unbounded copula density functions. We show that the estimator converges to infinity at the corner and we establish its relative convergence when the copula density is unbounded. Also, we provide the uniform strong consistency of the estimator on every compact in the interior region. We investigate the finite sample performance of the estimator via an extensive simulation study and we compare the Bernstein copula density estimator with other nonparametric methods. Finally, we consider an empirical application where the asymmetric dependence between international equity markets (US, Canada, UK, and France) is examined.

Variable selection in a flexible parametric mixture cure model with interval-censored data

In standard survival analysis, it is generally assumed that every individual will experience someday the event of interest. However, this is not always the case, as some individuals may not be susceptible to this event. Also, in medical studies, it is frequent that patients come to scheduled interviews and that the time to the event is only known to occur between two visits. That is, the data are interval‐censored with a cure fraction. Variable selection in such a setting is of outstanding interest. Covariates impacting the survival are not necessarily the same as those impacting the probability to experience the event. The objective of this paper is to develop a parametric but flexible statistical model to analyze data that are interval‐censored and include a fraction of cured individuals when the number of potential covariates may be large. We use the parametric mixture cure model with an accelerated failure time regression model for the survival, along with the extended generalized gamma for the error term. To overcome the issue of non‐stable and non‐continuous variable selection procedures, we extend the adaptive LASSO to our model. By means of simulation studies, we show good performance of our method and discuss the behavior of estimates with varying cure and censoring proportion. Lastly, our proposed method is illustrated with a real dataset studying the time until conversion to mild cognitive impairment, a possible precursor of Alzheimers disease.

Parametrically guided nonparametric density and hazard estimation with censored data

The parametrically guided kernel smoother is a promising nonparametric estimation approach that aims to reduce the bias of the classical kernel density estimator without increasing its variance. Theoretically, the estimator is unbiased if a correct parametric guide is used, which can never be achieved by the classical kernel estimator even with an optimal bandwidth. The estimator is generalized to the censored data case and used for density and hazard function estimation. The asymptotic properties of the proposed estimators are established and their performance is evaluated via finite sample simulations. The method is also applied to data coming from a study where the interest is in the time to return to drug use.

Tests for the equality of conditional variance functions in nonparametric regression

In this paper we are interested in checking whether the conditional variances are equal in k ≥ 2 location-scale models. Our procedure is fully nonparametric and is based on the comparison of the error distributions under the null hypothesis of equality of variances and without making use of this null hypothesis. We propose four test statistic based on empirical distribution functions (Kolmogorov-Smirnovand and Cramer-von Mises type test statistics) and two test statistics based on empirical characteristic functions. The limiting distributions of these six test statistics are established under the null hypothesis and under a local alternative. We show how to approximate the critical values under null using either an estimated version of the asymptotic null distribution or a bootstrap procedure. Simulation studies are conducted to assess the finite sample performance of the proposed tests. We also apply our tests to data on monthly expenditure.

Varying coefficient models having different smoothing variables with randomly censored data

The varying coefficient model is a useful alternative to the classical linear model, since the former model is much richer and more flexible than the latter. We propose estimators of the coefficient functions for the varying coefficient model in the case where different coefficient functions depend on different covariates and the response is sub ject to random right censoring. Since our model has an additive structure and requires multivariate smoothing we employ a smooth backfitting technique, that is known to be an effective way to avoid “the curse of dimensionality” in structured nonparametric models. The estimators are based on synthetic data obtained by an unbiased transformation. The asymptotic normality of the estimators is established, a simulation study illustrates the reliability of our estimators, and the estimation procedure is applied to data on drug abuse.