Nonparametric Inference for Copulas and Measures of Dependence Under Length-Biased Sampling and Informative Censoring

Length-biased data are often encountered in cross-sectional surveys and prevalent-cohort studies on disease durations. Under length-biased sampling subjects with longer disease durations have greater chance to be observed. As a result, covariate values linked to the longer survivors are favored by the sampling mechanism. When the sampled durations are also subject to right censoring, the censoring is informative. Modeling dependence structure without adjusting for these issues leads to biased results. In this article, we consider copulas for modeling dependence when the collected data are length-biased and account for both informative censoring and covariate bias that are naturally linked to length-biased sampling. We address nonparametric estimation of the bivariate distribution, copula function and its density, and Kendall and Spearman measures for right-censored length-biased data. The proposed estimator for the bivariate cdf is a Hadamard-differentiable functional of two MLEs (Kaplan–Meier and empirical cdf) and inherits their efficiency. Based on this estimator, we devise two estimators for copula function and a local-polynomial estimator for copula density that accounts for boundary bias. The limiting processes of the estimators are established by deriving their iid representations. As a by-product, we establish the oscillation behavior of the bivariate cdf estimator. In addition, we introduce estimators for Kendall and Spearman measures and study their weak convergence. The proposed method is applied to analyze a set of right-censored length-biased data on survival with dementia, collected as part of a nationwide study in Canada

Nonparametric Copula-Based Test for Conditional Independence with Applications to Granger Causality

This article proposes a new nonparametric test for conditional independence that can directly be applied to test for Granger causality. Based on the comparison of copula densities, the test is easy to implement because it does not involve a weighting function in the test statistic, and it can be applied in general settings since there is no restriction on the dimension of the time series data. In fact, to apply the test, only a bandwidth is needed for the nonparametric copula. We prove that the test statistic is asymptotically pivotal under the null hypothesis, establishes local power properties, and motivates the validity of the bootstrap technique that we use in finite sample settings. A simulation study illustrates the size and power properties of the test. We illustrate the practical relevance of our test by considering two empirical applications where we examine the Granger noncausality between financial variables. In a first application and contrary to the general findings in the literature, we provide evidence on two alternative mechanisms of nonlinear interaction between returns and volatilities: nonlinear leverage and volatility feedback effects. This can help better understand the well known asymmetric volatility phenomenon. In a second application, we investigate the Granger causality between stock index returns and trading volume. We find convincing evidence of linear and nonlinear feedback effects from stock returns to volume, but a weak evidence of nonlinear feedback effect from volume to stock returns

Nonparametric tests for conditional independence using conditional distributions

The concept of causality is naturally defined in terms of conditional distribution, however almost all the empirical works focus on causality in mean. This paper aims to propose a nonparametric statistic to test the conditional independence and Granger non-causality between two variables conditionally on another one. The test statistic is based on the comparison of conditional distribution functions using an L2 metric. We use Nadaraya–Watson method to estimate the conditional distribution functions. We establish the asymptotic size and power properties of the test statistic and we motivate the validity of the local bootstrap. We ran a simulation experiment to investigate the finite sample properties of the test and we illustrate its practical relevance by examining the Granger non-causality between S&P 500 Index returns and VIX volatility index. Contrary to the conventional t-test which is based on a linear mean-regression, we find that VIX index predicts excess returns both at short and long horizons

The metal elements traces dregs with the unstable fraction of the sediment of Sebou which risk?

In forms dissolved and/or fixed at the particles the metal elements traces can accumulate in the sedimentary zones and constitute important stocks of pollutants. These contaminants can be remaining in the dissolved phase and become biodisponible under the effect of the physicochemical conditions and could act as a long-term source of pollution. To understand the mobility and the reactivity of these stocks is an important issue for the management of the quality of the hydrosystèmes. The exchangeable or unstable fraction of the sediment corresponds to the metal ions being adsorbed on the surface of the particles constituting the sediment (clays, oxide iron…), or mobility and the biodisponibility are high in this fraction because of the weak electrostatic interactions. The goal of this study and to characterize the sediments of Sebou and Fès Rivers and to evaluate the risk of toxicity of the elements metal traces related to the particulate phase by the method of spared digestion. Metals in these fractions are assumed to be more available than metals associated with residual fractions. The results obtained show a strong mobilization of the elements related to the unstable fraction which reaches 100% for Pb and 70% for Cr. The sediments are composed of two mineralogical phases and they are also very rich in organic matter

Nonparametric estimation and inference for conditional density based Granger causality measures

We 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

Bernstein estimator for unbounded copula densities

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

A Nonparametric Copula Based Test for Conditional Independence with Applications to Granger Causality.

This article proposes a new nonparametric test for conditional independence that can directly be applied to test for Granger causality. Based on the comparison of copula densities, the test is easy to implement because it does not involve a weighting function in the test statistic, and it can be applied in general settings since there is no restriction on the dimension of the time series data. In fact, to apply the test, only a bandwidth is needed for the nonparametric copula. We prove that the test statistic is asymptotically pivotal under the null hypothesis, establishes local power properties, and motivates the validity of the bootstrap technique that we use in finite sample settings. A simulation study illustrates the size and power properties of the test. We illustrate the practical relevance of our test by considering two empirical applications where we examine the Granger noncausality between financial variables. In a first application and contrary to the general findings in the literature, we provide evidence on two alternative mechanisms of nonlinear interaction between returns and volatilities: nonlinear leverage and volatility feedback effects. This can help better understand the well known asymmetric volatility phenomenon. In a second application, we investigate the Granger causality between stock index returns and trading volume. We find convincing evidence of linear and nonlinear feedback effects from stock returns to volume, but a weak evidence of nonlinear feedback effect from volume to stock returns

Gamma Kernel Estimators for Density and Hazard Rate of Right-Censored Data

The nonparametric estimation for the density and hazard rate functions for right-censored data using the kernel smoothing techniques is considered. The “classical” fixed symmetric kernel type estimator of these functions performs well in the interior region, but it suffers from the problem of bias in the boundary region. Here, we propose new estimators based on the gamma kernels for the density and the hazard rate functions. The estimators are free of bias and achieve the optimal rate of convergence in terms of integrated mean squared error. The mean integrated squared error, the asymptotic normality, and the law of iterated logarithm are studied. A comparison of gamma estimators with the local linear estimator for the density function and with hazard rate estimator proposed by Müller and Wang (1994), which are free from boundary bias, is investigated by simulations