  The presence of confounding by high-dimensional variables complicates
estimation of the average effect of a point treatment. On the one hand, it
necessitates the use of variable selection strategies or more general
data-adaptive high-dimensional statistical methods. On the other hand, the use
of such techniques tends to result in biased estimators with a non-standard
asymptotic behaviour. Double-robust estimators are vital for offering a
resolution because they possess a so-called small bias property (Newey et al.,
2004). This means that their bias vanishes faster than the bias in the nuisance
parameter estimators when the relevant smoothing parameter goes to zero, making
their performance less sensitive to smoothing (Chernozhukov et al., 2016). This
property has been exploited to achieve valid (uniform) inference of the average
causal effect when data-adaptive estimators of the propensity score and
conditional outcome mean both converge to their respective truths at
sufficiently fast rate (e.g., van der Laan, 2014; Farrell, 2015; Belloni et
al., 2016). In this article, we extend this work in order to retain valid
(uniform) inference when one of these estimators does not converge to the
truth, regardless of which. This is done by generalising prior work for
low-dimensional settings by Vermeulen and Vansteelandt (2015) to incorporate
regularisation. The proposed penalised bias-reduced double-robust estimation
strategy exhibits promising performance in extensive simulation studies and a
data analysis, relative to competing proposals.
