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Improved nonparametric penalized maximum likelihood estimation for arbitrarily censored survival data

Nonparametric maximum likelihood estimation encompasses a group of classic methods to estimate distribution‐associated functions from potentially censored and truncated data, with extensive applications in survival analysis. These methods, including the Kaplan‐Meier estimator and Turnbull's method, often result in overfitting, especially when the sample size is small. We propose an improvement to these methods by applying kernel smoothing to their raw estimates, based on a BIC‐type loss function that balances the trade‐off between optimizing model fit and controlling model complexity. In the context of a longitudinal study with repeated observations, we detail our proposed smoothing procedure and optimization algorithm. With extensive simulation studies over multiple realistic scenarios, we demonstrate that our smoothing‐based procedure provides better overall accuracy in both survival function estimation and individual‐level time‐to‐event prediction (imputation) by reducing overfitting. Our smoothing procedure decreases the bias (discrepancy between the estimated and true simulated survival function) using interval‐censored data by up to 48% compared to the raw un‐smoothed estimate, with similar improvements of up to 34% and 23% in within‐sample and out‐of‐sample prediction, respectively. Our smoothing algorithm also demonstrates significant overall improvement across all three metrics when compared to a popular semiparametric B‐splines estimation method. Finally, we apply our method to real data on censored breast cancer diagnosis, which similarly shows improvement when compared to empirical survival estimates from uncensored data. We provide an R package, SISE, for implementing our penalized likelihood method.