Finite Sample Valid Inference via Calibrated Bootstrap

While widely used as a general method for uncertainty quantification, the bootstrap method encounters difficulties that raise concerns about its validity in practical applications. This paper introduces a new resampling-based method, termed calibrated bootstrap, designed to generate finite sample-valid parametric inference from a sample of size n. The central idea is to calibrate an m-out-of-n resampling scheme, where the calibration parameter m is determined against inferential pivotal quantities derived from the cumulative distribution functions of loss functions in parameter estimation. The method comprises two algorithms. The first, named resampling approximation (RA), employs a stochastic approximation algorithm to find the value of the calibration parameter m=m_ for a given in a manner that ensures the resulting m-out-of-n bootstrapped 1- confidence set is valid. The second algorithm, termed distributional resampling (DR), is developed to further select samples of bootstrapped estimates from the RA step when constructing 1- confidence sets for a range of values is of interest. The proposed method is illustrated and compared to existing methods using linear regression with and without L₁ penalty, within the context of a high-dimensional setting and a real-world data application. The paper concludes with remarks on a few open problems worthy of consideration.