Distribution-Free Prediction Intervals Under Covariate Shift, With an Application to Causal Inference

Owing to its appealing distribution-free feature, conformal inference has become a popular tool for constructing prediction intervals with a desired coverage rate. In scenarios involving covariate shift, where the shift function needs to be estimated from data, many existing methods resort to data-splitting techniques. However, these approaches often lead to wider intervals and less reliable coverage rates, especially when dealing with finite sample sizes. To address these challenges, we propose methods based on a pivotal quantity derived under a parametric working model and employ a resampling-based framework to approximate its distribution. The resampling-based approach can produce prediction intervals with a desired coverage rate without splitting the data and can be easily applied to causal inference settings where a shift in the covariate distribution can occur between treatment and control arms. Additionally, the proposed approaches enjoy a double robustness property and are adaptable to different prediction tasks. Our extensive numerical experiments demonstrate that, compared to existing methods, the proposed novel approaches can produce substantially shorter conformal prediction intervals with lower variability in the interval lengths while maintaining promising coverage rates and advantages in versatile usage.