An Extension of the d-Variate FGM Copula with Application

Abstract: We introduce an extended d-variate Farlie-Gumbel-Morgenstern (FGM) copula that incorporates additional parameters based on Legendre polynomials to enhance the representation of multivariate dependence structures. Within an i.i.d. framework, we derive closed-form estimators for these parameters and establish their unbiasedness, consistency, and asymptotic normality. A simulation study illustrates the finite-sample performance of the estimators. The model is applied to the Bearing dataset, previously studied by Ota and Kimura (2021) through a d-variate FGM copula and by Longla and Mous-Abou (2025) using an extended bivariate FGM copula. Our analysis shows that the classical d-variate FGM copula does not adequately represent the dependence in this dataset. Based on estimation results and model selection criteria, we propose a reduced version of the extended model as a more appropriate copula specification for the Bearing data.