Shape-constrained semiparametric maximum likelihood for backward time under an accelerated failure time model.

We investigate semiparametric maximum likelihood inference for backward (current-duration, backward-recurrence) times arising in cross-sectional (prevalent-cohort) sampling under an accelerated failure time (AFT) model for the underlying forward lifetime. A key observation is that, after a suitable transformation, the baseline density of backward times is intrinsically monotone non-increasing. Exploiting this shape constraint yields a profile likelihood estimator for the regression parameter β together with a nonparametric maximum likelihood estimator (NPMLE) of the monotone density via the Grenander estimator. Because the Grenander estimator is known to behave poorly near the origin-often causing instability in AFT parameter estimation-we additionally consider a log-concavity constraint on the backward density, which leads to substantially more stable estimation in practice. We present a rigorous formulation of the likelihood, provide sufficient conditions for identifiability, and establish consistency of the profile MLE. We further highlight the importance of restricting the parameter space for β to a compact set: this is essential for both identifiability and numerical stability, as the likelihood may otherwise be flat or ill-behaved. Finally, we propose a practical multi-start estimation procedure that integrates shape-restricted projection of the nonparametric component with derivative-free optimization of the parametric component.