A survey of kernel-type estimators for copula and their applications

Abstract

Abstract. Copulas have been widely used to model nonlinear dependence structure. Main applications of copulas include areas such as finance, insurance, hydrology, rainfall to name but a few. The flexibility of copula allows researchers to model dependence structure beyond Gaussian distribution. Basically, a copula is a function that couples multivariate distribution functions to their one-dimensional marginal distribution functions. In general, there are three methods to estimate copula. These are parametric, nonparametric, and semiparametric method. In this article we survey kernel-type estimators for copula such as mirror reflection kernel, beta kernel, transformation method and local likelihood transformation method. Then, we apply these kernel methods to three stock indexes in Asia. The results of our analysis suggest that, albeit variation in information criterion values, the local likelihood transformation method performs better than the other kernel methods.

Bivariate box plots based on quantile regression curves

“…If we do not have a parametric model for the copula (as in the preceding subsection), then we have to estimate the copula C and its partial derivative ∂ C. C can be estimated by using the empirical copula but, this estimator cannot be used to estimate the partial derivative. To this purpose it is better to use a kernel type estimator for C. A survey on the application of this kind of estimators to copula can be seen in [21]. The main problem is that the support of the copula is included in [ , ] while the kernel estimators do not have this property.…”

Distortion Representations of Multivariate Distributions

“…Note that both F and C can be estimated from the training sample by using empirical or kernel type estimators (see e.g. the survey in [25]).…”