  It is often of interest to assess whether a function-valued statistical
parameter, such as a density function or a mean regression function, is equal
to any function in a class of candidate null parameters. This can be framed as
a statistical inference problem where the target estimand is a scalar measure
of dissimilarity between the true function-valued parameter and the closest
function among all candidate null values. These estimands are typically defined
to be zero when the null holds and positive otherwise. While there is
well-established theory and methodology for performing efficient inference when
one assumes a parametric model for the function-valued parameter, methods for
inference in the nonparametric setting are limited. When the null holds, and
the target estimand resides at the boundary of the parameter space, existing
nonparametric estimators either achieve a non-standard limiting distribution or
a sub-optimal convergence rate, making inference challenging. In this work, we
propose a strategy for constructing nonparametric estimators with improved
asymptotic performance. Notably, our estimators converge at the parametric rate
at the boundary of the parameter space and also achieve a tractable null
limiting distribution. As illustrations, we discuss how this framework can be
applied to perform inference in nonparametric regression problems, and also to
perform nonparametric assessment of stochastic dependence.
