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 $\textit{calibrated bootstrap}$, 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 $\textit{resampling approximation}$ (RA), employs a $\textit{stochastic approximation}$ algorithm to find the value of the calibration parameter $m=m_\alpha$ for a given $\alpha$ in a manner that ensures the resulting $m$-out-of-$n$ bootstrapped $1-\alpha$ confidence set is valid. The second algorithm, termed $\textit{distributional resampling}$ (DR), is developed to further select samples of bootstrapped estimates from the RA step when constructing $1-\alpha$ confidence sets for a range of $\alpha$ values is of interest. The proposed method is illustrated and compared to existing methods using linear regression with and without $L_1$ 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.