This work provides a study of copula parameters estimators based on functions of Markov chains they generate. Copulas of interest are perturbations of the independence copula. We provide asymptotic distributions of maximum likelihood estimators and confidence intervals for parameters of several copula families introduced in Longla (Stat Pap 65:4331–4363, 2024). A set of moment-like estimators is proposed along with a multivariate asymptotic distribution. We investigate the particular case of Markov chains generated by sine copulas, sine-cosine copulas and the extended Farlie-Gumbel-Morgenstern (FGM) copula family. We provide evidence that our extension of the FGM copula does a better job at capturing the dependence in the Bearing Data Set . Some tests of independence of the sample are proposed. A simulation study is provided for the three examples of copula families that we propose from Type I Longla copulas. This simulation study proposes a comparative study of the introduced estimators, the maximum likelihood estimators and the robust estimators of Longla and Peligrad (J Stat Plan Inference 211:90–106, 2024). It showcases advantages of the proposed estimators.

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.