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It is often of interest to assess whether a function-valued statisticalparameter, such as a density function or a mean regression function, is equalto any function in a class of candidate null parameters. This can be framed asa statistical inference problem where the target estimand is a scalar measureof dissimilarity between the true function-valued parameter and the closestfunction among all candidate null values. These estimands are typically definedto be zero when the null holds and positive otherwise. While there iswell-established theory and methodology for performing efficient inference whenone assumes a parametric model for the function-valued parameter, methods forinference in the nonparametric setting are limited. When the null holds, andthe target estimand resides at the boundary of the parameter space, existingnonparametric estimators either achieve a non-standard limiting distribution ora sub-optimal convergence rate, making inference challenging. In this work, wepropose a strategy for constructing nonparametric estimators with improvedasymptotic performance. Notably, our estimators converge at the parametric rateat the boundary of the parameter space and also achieve a tractable nulllimiting distribution. As illustrations, we discuss how this framework can beapplied to perform inference in nonparametric regression problems, and also toperform nonparametric assessment of stochastic dependence.

Nonparametric inference on non-negative dissimilarity measures at the boundary of the parameter space