🤖 AI Summary
Classical d-variate FGM copulas inadequately capture complex, high-dimensional dependence structures inherent in bearing degradation data.
Method: We propose an extended FGM copula model based on Legendre polynomials, enabling explicit, closed-form estimation of high-dimensional FGM parameters for the first time. Under i.i.d. assumptions, we derive a parsimonious, interpretable, and computationally efficient version via simulation studies and model selection criteria (e.g., BIC).
Contribution/Results: We rigorously prove that the proposed estimator is unbiased, consistent, and asymptotically normal. Empirical evaluation demonstrates that classical FGM copulas severely underfit bearing data, whereas our model significantly improves dependence modeling accuracy. Crucially, it automatically identifies the optimal low-dimensional parameter structure—balancing theoretical rigor with practical applicability in reliability engineering and prognostics.
📝 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.