Title: Data-adaptive smoothing for optimal-rate estimation of possibly non-regular parameters,
Abstract: We consider nonparametric inference of finite dimensional, potentially
non-pathwise differentiable target parameters. In a nonparametric model, some
examples of such parameters that are always non pathwise differentiable target
parameters include probability density functions at a point, or regression
functions at a point. In causal inference, under appropriate causal
assumptions, mean counterfactual outcomes can be pathwise differentiable or
not, depending on the degree at which the positivity assumption holds.
In this paper, given a potentially non-pathwise differentiable target
parameter, we introduce a family of approximating parameters, that are pathwise
differentiable. This family is indexed by a scalar. In kernel regression or
density estimation for instance, a natural choice for such a family is obtained
by kernel smoothing and is indexed by the smoothing level. For the
counterfactual mean outcome, a possible approximating family is obtained
through truncation of the propensity score, and the truncation level then plays
the role of the index.
We propose a method to data-adaptively select the index in the family, so as
to optimize mean squared error. We prove an asymptotic normality result, which
allows us to derive confidence intervals. Under some conditions, our estimator
achieves an optimal mean squared error convergence rate. Confidence intervals
are data-adaptive and have almost optimal width.
A simulation study demonstrates the practical performance of our estimators
for the inference of a causal dose-response curve at a given treatment dose.