Detection of Heterogeneous Structures on the Gaussian Copula Model Using Projective Power Entropy

We discuss a parameter estimation problem for a Gaussian copula model under misspecification. Conventional estimators such as the maximum likelihood estimator (MLE) do not work well if the model is misspecified. We propose the estimator that minimizes the projective power entropy. We call it the -estimator, where denotes the power index. A feasible form of the projective power entropy is given that suites the Gaussian copula model. It is shown that the -estimator is robust against outliers. In addition the -estimator can appropriately detect a heterogeneous structure of the underlying distribution, even if the underlying distribution consists of some different copula components while a single Gaussian copula is used as a statistical model. We explore such an ability of the -estimator to detect the local structures in the comparison with the MLE. We also propose a fixed point algorithm to obtain the -estimator. The usefulness of the proposed methodology is demonstrated in numerical experiments. 1. Introduction Applications of copula models have been increasing in number in recent years. There are a variety of applications on finance, risk management [1] and multivariate time series analysis [2]. With copula models, the specification of the marginal distributions is parameterized separately from the dependence structure of the joint distribution. Hence, it gives a convenient way of the construction of flexible and more general multivariate distributions. As far as we know, there exist only a few works that are tackled with the identification and the statistical estimation of the mixture of copula models and most of them rely on MCMC algorithm. In this paper we focus on a misspecified Gaussian copula model. In other words, a sample follows a distribution mixed with different sources but a statistical model we fit is just a single Gaussian copula. It is very hard to construct multivariate copulas for three or more random variables [3], while the Gaussian is an exception. So we start with the Gaussian copula model, but later in Section 5 we will show that our method is closely related to -copula. As an example of misspecification, we consider that the underlying distribution is where is a mixing proportion and denotes the probability density function of the Gaussian copula with the correlation matrix parameter . We see that the MLE for almost surely converges to under the assumption (1), which means that the MLE fails to detect the structure of the underlying distribution. We make use of the -estimator [4, 5] that can be obtained via minimization of the