Title: Finite Sample Inference for Targeted Learning,
Abstract: The Highly-Adaptive-Lasso(HAL)-TMLE is an efficient estimator of a pathwise
differentiable parameter in a statistical model that at minimal (and possibly
only) assumes that the sectional variation norm of the true nuisance parameters
are finite. It relies on an initial estimator (HAL-MLE) of the nuisance
parameters by minimizing the empirical risk over the parameter space under the
constraint that sectional variation norm is bounded by a constant, where this
constant can be selected with cross-validation. In the formulation of the
HALMLE this sectional variation norm corresponds with the sum of absolute value
of coefficients for an indicator basis. Due to its reliance on machine
learning, statistical inference for the TMLE has been based on its normal limit
distribution, thereby potentially ignoring a large second order remainder in
finite samples.
In this article, we present four methods for construction of a finite sample
0.95-confidence interval that use the nonparametric bootstrap to estimate the
finite sample distribution of the HAL-TMLE or a conservative distribution
dominating the true finite sample distribution. We prove that it consistently
estimates the optimal normal limit distribution, while its approximation error
is driven by the performance of the bootstrap for a well behaved empirical
process. We demonstrate our general inferential methods for 1) nonparametric
estimation of the average treatment effect based on observing on each unit a
covariate vector, binary treatment, and outcome, and for 2) nonparametric
estimation of the integral of the square of the multivariate density of the
data distribution.