Penalized EM algorithm and copula skeptic graphical models for inferring networks for mixed variables

In this article, we consider the problem of reconstructing networks for continuous, binary, count and discrete ordinal variables by estimating sparse precision matrix in Gaussian copula graphical models. We propose two approaches: $\ell_1$ penalized extended rank likelihood with Monte Carlo Expectation-Maximization algorithm (copula EM glasso) and copula skeptic with pair-wise copula estimation for copula Gaussian graphical models. The proposed approaches help to infer networks arising from nonnormal and mixed variables. We demonstrate the performance of our methods through simulation studies and analysis of breast cancer genomic and clinical data and maize genetics data

Improving Upon the Marginal Empirical Distribution Functions when the Copula is Known

At the heart of the copula methodology in statistics is the idea of separating marginal distributions from the dependence structure. However, as shown in this paper, this separation is not to be taken for granted: in the model where the copula is known and the marginal distributions are completely unknown, the empirical distribution functions are semiparametrically efficient if and only if the copula is the independence copula. Incorporating the knowledge of the copula into a nonparametric likelihood yields an estimation procedure which by simulations is shown to outperform the empirical distribution functions, the amount of improvement depending on the copula. Although the known-copula model is arguably artificial, it provides an instructive stepping stone to the more general model of a parametrically specified copula and arbitrary margins.independence copula;nonparametric maximum likelihood estimator;score function;semiparametric efficiency;tangent space

Copula estimation for nonsynchronous financial data

Copula is a powerful tool to model multivariate data. We propose the modelling of intraday financial returns of multiple assets through copula. The problem originates due to the asynchronous nature of intraday financial data. We propose a consistent estimator of the correlation coefficient in case of Elliptical copula and show that the plug-in copula estimator is uniformly convergent. For non-elliptical copulas, we capture the dependence through Kendall's Tau. We demonstrate underestimation of the copula parameter and use a quadratic model to propose an improved estimator. In simulations, the proposed estimator reduces the bias significantly for a general class of copulas. We apply the proposed methods to real data of several stock prices

Impact of non-stationarity on estimating and modeling empirical copulas of daily stock returns

All too often measuring statistical dependencies between financial time series is reduced to a linear correlation coefficient. However this may not capture all facets of reality. We study empirical dependencies of daily stock returns by their pairwise copulas. Here we investigate particularly to which extent the non-stationarity of financial time series affects both the estimation and the modeling of empirical copulas. We estimate empirical copulas from the non-stationary, original return time series and stationary, locally normalized ones. Thereby we are able to explore the empirical dependence structure on two different scales: a global and a local one. Additionally the asymmetry of the empirical copulas is emphasized as a fundamental characteristic. We compare our empirical findings with a single Gaussian copula, with a correlation-weighted average of Gaussian copulas, with the K-copula directly addressing the non-stationarity of dependencies as a model parameter, and with the skewed Student's t-copula. The K-copula covers the empirical dependence structure on the local scale most adequately, whereas the skewed Student's t-copula best captures the asymmetry of the empirical copula on the global scale.Comment: 20 page

Gaussian Process Conditional Copulas with Applications to Financial Time Series

The estimation of dependencies between multiple variables is a central problem in the analysis of financial time series. A common approach is to express these dependencies in terms of a copula function. Typically the copula function is assumed to be constant but this may be inaccurate when there are covariates that could have a large influence on the dependence structure of the data. To account for this, a Bayesian framework for the estimation of conditional copulas is proposed. In this framework the parameters of a copula are non-linearly related to some arbitrary conditioning variables. We evaluate the ability of our method to predict time-varying dependencies on several equities and currencies and observe consistent performance gains compared to static copula models and other time-varying copula methods