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August 21, 2025

Some results on point estimation of the association parameter of a bivariate Frank copula

TPThi-Yen-Anh PhamVietnam National University Ho Chi Minh City THTai Thien HuynhHo Chi Minh City University of Science NPNabendu PalTon Duc Thang University

Key Points

The results show that method of moments estimators perform well near zero while maintaining low bias, differing from maximum likelihood estimators.
In terms of mean squared error, maximum likelihood estimators exhibit superior overall performance compared to other methods.
The study investigates biases and errors in point estimation methods, highlighting their implications for financial applications using bivariate frank copulas.
These findings offer valuable insights into the behaviors of different point estimators, indicating a need for further exploration in statistical inference frameworks.

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, primarily 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), standardized bias (SB) and standardized MSE (SMSE). 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, our results, which have clearly demonstrated some interesting patterns, do not match with those reported by Genest (in Citation1987) and Susam (in Citation2020). Nevertheless, this study complements Genest's expository work, and provides some interesting insights into the behaviors of three point estimators including the MLE whose asymptotic behavior holds quite well, as we have found, for n≥75.

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