Nonparametric inference via bootstrapping the debiased estimator

Abstract

In this paper, we propose to construct confidence bands by bootstrapping the debiased kernel density estimator (for density estimation) and the debiased local polynomial regression estimator (for regression analysis). The idea of using a debiased estimator was first introduced in Calonico et al. (2015), where they construct a confidence interval of the density function (and regression function) at a given point by explicitly estimating stochastic variations. We extend their ideas and propose a bootstrap approach for constructing confidence bands that is uniform for every point in the support. We prove that the resulting bootstrap confidence band is asymptotically valid and is compatible with most tuning parameter selection approaches, such as the rule of thumb and cross-validation. We further generalize our method to confidence sets of density level sets and inverse regression problems. Simulation studies confirm the validity of the proposed confidence bands/sets.

Nonparametric confidence regions for level sets: Statistical properties and geometry

“…Bootstrap confidence regions based on vertical variation of the kernel density estimate have been constructed in Mammen and Polonik (2013) and in Chen et al (2017). They are based on a bootstrap approximation of quantiles of statistics of the form…”

“…Nevertheless, the topology (homology) of the confidence regions will in general be different from the one of the target contour, and the normal compatibility assumption (e.g. see Chazal et al 2007) used in Chen et al (2017) for the construction of horizontal variation based confidence regions will be violated in such cases.…”

“…where f l denotes a kernel density estimator using bandwidth l, which can be different from h. See Chen (2017) for a recent study of the de-biased estimator.…”

“…Both of these approaches are being discussed in our work. For further relevant recent literature see Chen (2017) and Calonico et al (2018a).…”

“…Bootstrap confidence regions for a density superlevel set and/or isosurface based on vertical variation can be found in Mammen and Polonik (2013) and Chen et al (2017). The latter also constructed a confidence set based on horizontal variation, and such constructions can also be found in Chen (2017). The confidence sets proposed here are compared to these methods.…”