  We propose a two-stage estimation procedure for a copula-based model with
semi-competing risks data, where the non-terminal event is subject to dependent
censoring by the terminal event, and both events are subject to independent
censoring. Under a copula-based model, the marginal survival functions of
individual event times are specified by semiparametric transformation models,
and the dependence between the bivariate event times is specified by a
parametric copula function. For the estimation procedure, in the first stage,
the parameters associated with the marginal of the terminal event are estimated
only using the corresponding observed outcomes, and in the second stage, the
marginal parameters for the non-terminal event time and the copula parameter
are estimated via maximizing a pseudo-likelihood function based on the joint
distribution of the bivariate event times. We derived the asymptotic properties
of the proposed estimator and provided an analytic variance estimator for
inference. Through simulation studies, we showed that our approach leads to
consistent estimates with less computational cost and more robustness compared
to the one-stage procedure developed in Chen (2012), where all parameters were
estimated simultaneously. In addition, our approach demonstrates more desirable
finite-sample performances over another existing two-stage estimation method
proposed in Zhu et al. (2021).
