Asymptotic efficiency of targeted maximum likelihood estimators (TMLE) of target features of the data distribution relies on a a second order remainder being asymptotically negligible. In previous work we proposed a nonparametric MLE termed Highly Adaptive Lasso (HAL) which parametrizes the relevant functional of the data distribution in terms of a multivariate real valued cadlag function that is assumed to have finite variation norm. We showed that the HAL-MLE converges in Kullback-Leibler dissimilarity at a rate n-1/3 up till logn factors. Therefore, by using HAL as initial density estimator in the TMLE, the resulting HAL-TMLE is an asymptotically efficient estimator only assuming that the relevant nuisance functions of the data density are cadlag and have finite variation norm. However, in finite samples, the second order remainder can dominate the sampling distribution so that inference based on asymptotic normality would be anti-conservative. In this article we propose a new higher order TMLE, generalizing the regular first order TMLE. We prove that it satisfies an exact linear expansion, in terms of efficient influence functions of sequentially defined higher order fluctuations of the target parameter, with a remainder that is a k+1th order remainder. As a consequence, this k-th order TMLE allows statistical inference only relying on the k+1th order remainder being negligible. We also provide a rationale for the higher order TMLE that it will be superior to the first order TMLE by (iteratively) locally minimizing the exact finite sample remainder of the first order TMLE. The second order TMLE is demonstrated for nonparametric estimation of the integrated squared density and for the treatment specific mean outcome. We also provide an initial simulation study for the second order TMLE of the treatment specific mean confirming the theoretical analysis.

翻译：在先前的工作中,我们提出了一个非对称 MLE 称为高度适应性拉索(HAL), 假设数据密度的相关微调功能具有一定的变异性规范。我们显示, HAL-MLE 会在 Kullback- Leibild 的分布中以n-1/3 的速率和日志因数相悖。因此,通过在TMLE中将 HAL 用作初始密度比值的测算器,产生的 HAL-TMLE 是一个非对称 MLE (HAL) 的高效测算器,它只能假设数据密度的相关微调功能是封闭的,并且具有一定的变异规范。然而,在有限的样本中,第二顺序可以控制取样分布,这样基于不均度的推论将具有反调性。因此,在本文章中,我们提出一个新的更高顺序的测算值为 IMLE+ 的直径直值比值, 也能够使常规的直径顺序得到常规的测序。