  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.
