Some Results on Point Estimation of the Association Parameter of a Bivariate Frank Copula

TL;DR

This paper comprehensively evaluates three point estimators for the association parameter of a bivariate Frank Copula, highlighting their bias, MSE, and performance differences, especially near zero and for larger sample sizes.

Contribution

It provides a detailed comparison of three estimators for the Frank Copula's association parameter, including bias and MSE analysis, extending previous work by Genest (1987).

Findings

MMEs perform well near zero compared to MLE

MLE has the best overall performance

Asymptotic behavior of MLE is reliable for n ≥ 75

Abstract

This work deals with estimation of the association parameter of a bivariate Frank Copula in a comprehensive way. Even though Frank Copula is a member of Archimedean class of copulas, and has been widely used in finance, relatively little attention has been paid to its association parameter from a statistical inferential point of view. Most of the existing works which have used Frank Copula have focused on estimating the parameter computationally, and then proceeded with its application in the applied fields, mostly in finance. Here, in this investigation, we have looked at the point estimation of the association parameter in a comprehensive manner, and studied three estimators in terms of bias, mean squared error (MSE), relative bias and relative MSE. It has been noted that in the neighborhood of zero, the method of moment estimators (MMEs) do perform well compared to the maximum likelihood estimator (MLE), even though the latter has the best overall performance. Further, in terms of bias, MMEs and MLE have opposite behavior. However, some of our results do not match with those reported by Genest (1987) \cite{Genest1987}. Nevertheless, this study complements Genest's (1987)\cite{Genest1987} expository work, and provides some interesting insights into the behaviors of three point estimators including the MLE whose asymptotic behavior holds pretty well, as we have found, for $n\ge 75$.