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 recently employed by Calonico et al. (2018b) to construct a confidence interval of the density function (and regression function) at a given point by explicitly estimating stochastic variations. We extend their ideas of using the debiased estimator and further propose a bootstrap approach for constructing simultaneous confidence bands. This modified method has an advantage that we can easily choose the smoothing bandwidth from conventional bandwidth selectors and the confidence band will be asymptotically valid. We prove the validity of the bootstrap confidence band and generalize it to density level sets and inverse regression problems. Simulation studies confirm the validity of the proposed confidence bands/sets. We apply our approach to an Astronomy dataset to show its applicability.
Highlights
In nonparametric statistics, how to construct a confidence band has been a central research topic for several decades
We introduce a simple approach to constructing confidence bands for both density and regression functions by bootstrapping a debiased estimator, which can be viewed as a synthesis of both the debiased and the undersmoothing methods
We introduce the debiased estimator for the local polynomial regression (Fan and Gijbels, 1996; Wasserman, 2006)
Summary
How to construct a confidence band has been a central research topic for several decades. We introduce a simple approach to constructing confidence bands for both density and regression functions by bootstrapping a debiased estimator, which can be viewed as a synthesis of both the debiased and the undersmoothing methods. Our method is based on the debiased estimator proposed in Calonico et al (2018b), where the authors propose a confidence interval of a fixed point using an explicit estimation of the errors They consider univariate density and their approach is only valid for a given point, which limits the applicability. This leads to a simple but elegant approach of constructing a valid confidence band with a uniform.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have