Copula-Based Bivariate Modified Fréchet–Exponential Distributions: Construction, Properties, and Applications

The classical exponential model, despite its flexibility, fails to describe data with non-constant failure or between-event dependency. To overcome this limitation, two new bivariate lifetime distributions are introduced in this paper. The Farlie–Gumbel–Morgenstern (FGM)-based and Ali–Mikhail–Haq (AMH)-based modified Fréchet–exponential (MFE) models, by embedding the flexible MEF margin in the FGM and AMH copulas. The resulting distributions accommodate a wide range of positive or negative dependence while retaining analytical traceability. Closed-form expressions for the joint and marginal density, survival, hazard, and reliability functions are derived, together with product moments and moment-generating functions. Unknown parameters are estimated through the maximum likelihood estimation (MLE) and inference functions for margins (IFM) methods, with asymptotic confidence intervals provided for these parameters. An extensive Monte Carlo simulation quantifies the bias, mean squared error, and interval coverage, indicating that IFM retains efficiency while reducing computational complexity for moderate sample sizes. The models are validated using two real datasets, from the medical sector regarding the infection recurrence times of 30 kidney patients undergoing peritoneal dialysis, and from the economic sector regarding the growth of the gross domestic product (GDP). Overall, the proposed copula-linked MFE distributions provide a powerful and economical framework for survival analysis, reliability, and economic studies.