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'.
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 Alzheimer's disease. © 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.
The prediction reliability is of primary concern in many clinical studies when the objective is to develop new predictive models or improve existing risk scores. In fact, before using a model in any clinical decision making, it is very important to check its ability to discriminate between subjects who are at risk of, for example, developing certain disease in a near future from those who will not. To that end, the time-dependent receiver operating characteristic (ROC) curve is the most commonly used method in practice. Several approaches have been proposed in the literature to estimate the ROC nonparametrically in the context of survival data. But, except one recent approach, all the existing methods provide a nonsmooth ROC estimator whereas, by definition, the ROC curve is smooth. In this article we propose and study a new nonparametric smooth ROC estimator based on a weighted kernel smoother. More precisely, our approach relies on a well-known kernel method used to estimate cumulative distribution functions of random variables with bounded supports. We derived some asymptotic properties for the proposed estimator. As bandwidth is the main parameter to be set, we present and study different methods to appropriately select one. A simulation study is conducted, under different scenarios, to prove the consistency of the proposed method and to compare its finite sample performance with a competitor. The results show that the proposed method performs better and appear to be quite robust to bandwidth choice. As for inference purposes, our results also reveal the good performances of a proposed nonparametric bootstrap procedure. Furthermore, we illustrate the method using a real data example.
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 companies.
Publisher's copyright statement: NOTICE: this is the author's version of a work that was accepted for publication in Journal of Econometrics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be re ected in this document. Changes may have been made to this work since it was submitted for publication. A de nitive version was subsequently published in Journal of Econometrics, 180, 2, June 2014, 10.1016/j.jeconom.2014.03.001. Additional information:Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. ABSTRACTWe propose a nonparametric estimation and inference for conditional density based Granger causality measures that quantify linear and nonlinear Granger causalities. We first show how to write the causality measures in terms of copula densities. Thereafter, we suggest consistent estimators for these measures based on a consistent nonparametric estimator of copula densities. Furthermore, we establish the asymptotic normality of these nonparametric estimators and discuss the validity of a local smoothed bootstrap that we use in finite sample settings to compute a bootstrap bias-corrected estimator and to perform statistical tests. A Monte Carlo simulation study reveals that the bootstrap bias-corrected estimator behaves well and the corresponding test has quite good finite sample size and power properties for a variety of typical data generating processes and different sample sizes. Finally, two empirical applications are considered to illustrate the practical relevance of nonparametric causality measures.JEL Classification: C12; C14; C15; C19; G1; G12; E3; E4.
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