A Wald-type variance estimation for the nonparametric distribution estimators for doubly censored data

We discuss the variance estimation for the nonparametric distribution estimator for doubly censored data. We first provide another view of Kuhn–Tucker’s conditions to construct the profile likelihood, and lead a Newton–Raphson algorithm as an optimization technique unlike the EM algorithm. The main proposal is an iteration-free Wald-type variance estimate based on the chain rule of differentiating conditions to construct the profile likelihood, which generalizes the variance formula in only right- or left-censored data. In this estimation procedure, we overcome some difficulties caused in directly applying Turnbull’s formula to large samples and avoid a load with computationally heavy iterations, such as solving the Fredholm equations, computing the profile likelihood ratio or using the bootstrap. Also, we establish the consistency of the formulated Wald-type variance estimator. In addition, simulation studies are performed to investigate the properties of the Wald-type variance estimates in finite samples in comparison with those from the profile likelihood ratio.