Probit transformation for nonparametric kernel estimation of the copula density

Abstract

Copula modelling has become ubiquitous in modern statistics. Here, the problem of nonparametrically estimating a copula density is addressed. Arguably the most popular nonparametric density estimator, the kernel estimator is not suitable for the unit-square-supported copula densities, mainly because it is heavily affected by boundary bias issues. In addition, most common copulas admit unbounded densities, and kernel methods are not consistent in that case. In this paper, a kernel-type copula density estimator is proposed. It is based on the idea of transforming the uniform marginals of the copula density into normal distributions via the probit function, estimating the density in the transformed domain, which can be accomplished without boundary problems, and obtaining an estimate of the copula density through back-transformation. Although natural, a raw application of this procedure was, however, seen not to perform very well in the earlier literature. Here, it is shown that, if combined with local likelihood density estimation methods, the idea yields very good and easy to implement estimators, fixing boundary issues in a natural way and able to cope with unbounded copula densities. The asymptotic properties of the suggested estimators are derived, and a practical way of selecting the crucially important smoothing parameters is devised. Finally, extensive simulation studies and a real data analysis evidence their excellent performance compared to their main competitors.

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

“…Extension of transformation method is proposed by [18] by fitting polynomial locally to the log-density (TLL1) and by quadratic polynomials (TLL2). Let Z i = (U i , V i ) and let ∆ kn be Euclidean distance between (x, y) and the kth closest observation between (…”

“…Various applications of the kernel-type estimators in previous section can be found in [11], [18], [12] and [17]. In this section we apply the kernel estimators to three weekly log return of stock indexes in Asia (STI, HSI and AORD) for period 3 January 2000 to 7 March 2016.…”

Nonparametric universal copula modeling

“…In addition, these parametric classes of copula models are notoriously less flexible when it comes to capturing real‐world complex dependency structures. As a remedy, several nonparametric kernel‐based procedures have been developed in recent times . X , Y both discrete: Copula estimation is far more challenging for discrete margins that include binary, ordinal categorical, and count data.…”

“…We perform numerical comparisons with state‐of‐the‐art nonparametric copula density estimation methods, implemented in the R package kdecopula: Probit transformation estimator with log‐quadratic local likelihood estimation and nearest‐neighbor bandwidths . The classical mirror‐reflection estimator proposed by Gijbels and Mielniczuk .…”

“…We restrict ourselves to continuous margins, as the competing methods are not generalizable for mixed data‐type. Our numerical setting closely follows, section 5. We simulate B =250 independent random copula‐samples of size n =1000 from the following copula distributions: Gaussian copula with parameter ρ=0.70; Student's t‐copula with 5 degrees of freedom, with ρ=−0.30; Frank copula with parameter θ=6,−2. Plackett Copula with parameter θ=6,0.10. Clayton copula with parameter θ=3,−0.50. Ali‐Mikhail‐Haq copula with parameter θ=0.85,−0.85. Joe copula with parameter θ=1.5. Gumbel copula with parameter θ=1.5. …”