The Efficiency of the Hotelling’s $$\boldsymbol{T^{2}}$$ Control Chart Performance using Bivariate Copulas: A Monte Carlo Simulation Approach with Kendall’s Tau Dependence

In this study, we explore the combination of bivariate copulas, drawing from the Gumbel, Clayton, FGM, Frank, AMH, normal, and Joe copulas. This study accentuates the asymmetric characteristics of Gumbel, Clayton, Frank, and Joe copulas, setting them apart through their unique parameter spaces. We also focus significantly on the potential of bivariate copulas in enhancing Hotelling’s $$T^{2}$$ control chart, particularly leveraging the symmetric properties of FGM, AMH, and normal copulas. A crucial aspect of our research is the emphasis on the prominence of the FGM family within substantial parametric copula families. Our methodology explores a multivariate class rooted in the combination of bivariate copulas, particularly involving AMH and FGM. Our analytical encompasses a rich variety of copula combinations, culminating in 21 combined copulas. By using Kendall’s tau ( $$\tau$$ ) values, the dependency strengths is analyzed at levels of 0.0, 0.2 as weak, 0.5 as moderate, and 0.9 as strong, aiding in the effective determination of dependence between variables within a normal distribution framework characterized by parameters $$\mu=0$$ and $$\sigma^{2}=1$$ . Utilizing Monte Carlo simulation, we evaluated the performance dynamics of various copula combinations within Hotelling’s $$T^{2}$$ control chart, focusing on average run length (ARL) as the primary evaluation metric. The findings of this study corroborate the efficacy of the bivariate copulas methodology in fitting Hotelling’s $$T^{2}$$ control chart, indicating a marginal difference in performance across different dependence levels, especially at significant shifts ( $$\delta\geq 2$$ ). By analyzing observation dependencies to identify the optimal copula fitting for a given dataset, this research shows the vital role of copulas in examining the dependency dynamics of multivariate systems.