Estimation and Inference for Nonparametric Expected Shortfall Regression over RKHS

Expected shortfall (ES) has emerged as an important metric for characterizing the tail behavior of a random outcome, specifically associated with rarer events that entail severe consequences. In climate science, the threats of flooding and heatwaves loom large, impacting natural environments and human communities. In actuarial studies, a key observation in modeling insurance claim sizes is that features exhibit distinct effects in explaining small and large claims. This paper concerns nonparametric expected shortfall regression as a class of statistical methods for tail learning. These methods directly target upper/lower tail averages and will empower practitioners to address complex questions that are beyond the reach of mean regression-based approaches. Using kernel ridge regression, we introduce a two-step nonparametric ES estimator that involves a plugged-in quantile function estimate without sample-splitting. We provide non-asymptotic estimation and Gaussian approximation error bounds, depending explicitly on the effective dimension, sample size, regularization parameters, and quantile estimation error. To construct pointwise confidence bands, we propose a fast multiplier bootstrap procedure and establish its validity. We demonstrate the finite-sample performance of the proposed methods through numerical experiments and an empirical study aimed at examining the heterogeneous effects of different air pollutants and meteorological factors on average and high PM2.5 concentration.