Estimation of the FGM Bivariate Exponentiated Weibull Distribution under Progressive Type-II Censoring

Abstract

The bivariate Exponentiated Weibull distribution is a flexible lifetime model for paired survival data. Coupling Exponentiated Weibull margins through the Farlie–Gumbel–Morgenstern (FGM) copula yields the FGM bivariate Exponentiated Weibull (FGMBEW) distribution, which is appropriate for bivariate lifetime data exhibiting weak dependence. Estimation for the FGMBEW distribution has been studied for complete data, but not under progressive censoring, which arises routinely in life-testing and reliability experiments. This paper develops maximum likelihood (ML) and inference functions for margins (IFM) estimation of the FGMBEW distribution under Progressive Type-II censoring. The censoring mechanism is specified explicitly, the full and two-stage likelihoods are derived, and closed-form score equations are given. A Monte Carlo study with 1000 replications, three contrasting removal patterns and six sample sizes shows that bias and mean squared error decrease monotonically in the sample size for every parameter and both methods. The two estimators are close in accuracy: mean squared error ratios lie between 0.79 and 1.47, with IFM slightly preferable for the dependence parameter  under every scheme and ML preferable for  when removals are concentrated late. IFM is 4.2 to 5.2 times faster than joint ML. Confidence intervals based on the observed information for the ML estimator under-cover substantially in small samples (0.645 against a nominal 0.95 at  under late removals), whereas the two-stage intervals are close to or slightly above nominal throughout. The methods are illustrated on the kidney infection recurrence data of McGilchrist and Aisbett (1991), for which Kendall’s  and Spearman’s  fall inside the FGM admissible ranges. A Cramér–von Mises test with a parametric bootstrap does not reject the FGM copula, but the profile likelihood for  is nearly flat, so the dependence parameter is only weakly determined at the observed sample size.