Analysis of left-truncated and interval-censored survival data is challenging, particularly when the failure time and observation process are dependent. Existing methods, including Sun et al. (2023), model dependency via copulas, which may be restrictive in the presence of complex interval-censoring mechanisms. To address this gap, we propose the first shared frailty model specifically designed for left-truncated, interval-censored data, capturing heterogeneity and dependency between the failure time and observation processes. A sieve maximum likelihood approach is developed, using I-splines and M-splines to approximate the unknown baseline hazard and examination intensity functions. The asymptotic properties of the estimators are established, and an extensive simulation study demonstrates that the method provides consistent, efficient, and robust parameter estimates under a variety of scenarios. The approach is illustrated through a real data application to the AIDS cohort study, highlighting its ability to account for left truncation, interval censoring, and dependent observation process.
