A New Kernel Estimator of Copulas Based on Beta Quantile Transformations

Catalina Bolancé,Carlos Alberto Acuña

doi:10.3390/math9101078

Abstract

A copula is a multivariate cumulative distribution function with marginal distributions Uniform(0,1). For this reason, a classical kernel estimator does not work and this estimator needs to be corrected at boundaries, which increases the difficulty of the estimation and, in practice, the bias boundary correction might not provide the desired improvement. A quantile transformation of marginals is a way to improve the classical kernel approach. This paper shows a Beta quantile transformation to be optimal and analyses a kernel estimator based on this transformation. Furthermore, the basic properties that allow the new estimator to be used for inference on extreme value copulas are tested. The results of a simulation study show how the new nonparametric estimator improves alternative kernel estimators of copulas. We illustrate our proposal with a financial risk data analysis.

Highlights

Based on the kernel method and transformations, we present a new nonparametric estimator of a multivariate copula that improves the empirical copula and the most prominent kernel estimators
We summarise the results of our simulation study, we aim to evaluate the finite sample properties of our Beta transformed kernel estimator in (12), and compare it with the empirical copula, with the classical kernel estimator in (5) and with the Gaussian transformed kernel estimator in (7)
We show two types of results; in the former, the errors between the estimations and true copulas are compared and, in the latter, the differences between the extreme value copula tests obtained with the empirical copula and with the Beta transformed kernel estimator are analysed

Summary

Introduction

Based on the kernel method and transformations, we present a new nonparametric estimator of a multivariate copula that improves the empirical copula and the most prominent kernel estimators (see reference [1], for a detailed review). The copula model allows us to represent the dependence structure of a multivariate random vector of continuous variables X = X1, ..., XJ , which combines with marginal distributions to give the multivariate distribution This idea was established in the fundamental theorem proposed by Sklar [2]. A nonparametric estimation of a copula can be obtained whose results can be used for estimating joint probabilities or for testing the adequacy of a copula family, for example, the extreme value copula family. In this paper, these two aims of our new nonparametric estimator are analysed through a simulation study

Objectives

Results

Conclusion