On Exact Confidence Intervals for Nonparametric Regression

In nonparametric regression, constructing valid and efficient confidence intervals remains as a challenging problem. This is particularly true when the underlying function has a wide range of unknown degrees of smoothness. Here, we consider γ-H ̈older continuous (including Lipschitz) functions with 0 < γ ≤ 1. For nonparametric regression with independent Gaussian errors, we provide a method of constructing valid and efficient point-wise confidence intervals. The proposed method is based on the Inferential Models framework of Martin and Liu (2015). Complications due to the continuity constraints make it challenging to construct valid and effi- cient predictive random sets. For this, a new technique, called partial conditioning, is developed to take the advantages of both conditional and marginal inferential models. Theoretical re- sults are established to show that resulting confidence intervals are exact in the sense with right coverage. Numerical methods are provided for illustrative numerical examples, including Brownian-like continuous functions that have their H ̈odler exponent 0 < γ < 12 . The asymptotic efficiency of the proposed method is also investigated empirically.
References: Ryan Martin and Chuanhai Liu. Inferential models: reasoning with uncertainty, volume 145. CRC Press, 2015.