Nonparametric maximum likelihood estimators (MLEs) in inverse problems often have non-normal limit distributions, like Chernoff's distribution. However, if one considers smooth functionals of the model, with corresponding functionals of the MLE, one gets normal limit distributions and faster rates of convergence. We demonstrate this for a model for the incubation time of a disease. The usual approach in the latter models is to use parametric distributions, like Weibull and gamma distributions, which leads to inconsistent estimators. Smoothed bootstrap methods are discussed for constructing confidence intervals. The classical bootstrap, based on the nonparametric MLE itself, has been proved to be inconsistent in this situation.

翻译：逆问题中的非参数最大似然估计通常具有非正态极限分布，例如切尔诺夫分布。然而，若考虑模型的平滑泛函及其对应的最大似然估计泛函，则可获得正态极限分布及更快的收敛速度。我们针对疾病潜伏期模型验证了这一点。传统方法通常采用参数化分布（如威布尔分布和伽马分布），但这会导致不一致的估计量。本文讨论了基于平滑自助法构造置信区间的方法。在此情形下，基于非参数最大似然估计本身的经典自助法已被证明是不一致的。