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 interval censoring models and a model for the incubation time of Covid-19. 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 choosing a bandwidth and constructing confidence intervals. The classical bootstrap, based on the nonparametric MLE itself, has been proved to be inconsistent in this situation.
翻译:在反向问题中,非对称最大概率估测器(MLEs)往往有非正常的极限分布,如Chernoff的分布。然而,如果考虑到模型的顺利功能,加上MLE的相应功能,人们就会得到正常的极限分布和更快的趋同率。我们用间隙审查模型和Covid-19孵化时间模型来证明这一点。后一种模型的通常做法是使用参数分布,如Weibull和伽马分布,这导致测算器不一致。在选择带宽和构建信任间隔时,人们会讨论光滑靴套式方法。在这种情况下,基于非参数MLE本身的经典靴套被证明是不一致的。