Inference using recall-based data from log-normal distribution

In cross-sectional and survey studies, time-to-event data is often collected retrospectively through historical records or individual responses, resulting in recall-based data. Assuming a log-normal distribution for the lifetime data, we developed an estimation procedure tailored to this context. Point estimates were obtained using the quasi-Newton and expectation-maximization algorithms, while the standard errors of the maximum likelihood estimators were calculated based on their asymptotic properties. To address parameter uncertainty, we employed Bayesian methods with conjugate prior distributions. A simulation study was conducted to evaluate the performance of proposed estimators, considering varying levels of missing data across different sample sizes. Finally, the proposed methods were applied to estimate the age at menarche using cross-sectional data.