A New Kernel Estimator of Copulas Based on Beta Quantile Transformations

• Main Authors: Catalina Bolancé, Carlos Alberto Acuña
• Format: Article
• Language: English
• Published: MDPI AG 2021-05-01
• Series: Mathematics
• Subjects: nonparametric copula; kernel estimation; Beta transformation; extreme value copula
• Online Access: https://www.mdpi.com/2227-7390/9/10/1078

A copula is a multivariate cumulative distribution function with marginal distributions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>U</mi><mi>n</mi><mi>i</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>. 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.