Variance-Reduced Risk Inference in Semi-Supervised Settings

In the estimating equation framework, this paper develops a variance-reduced estimation procedure when next to a short sequence of interest another longer auxiliary sequence is available. The proposed method does not require modeling and inferring the dependence between the short and long sequences. We apply the proposed method to develop a novel variance-reduced estimator for three popular risk measures: Value-at-Risk, Expected Shortfall, and Expectile. A simulation study confirms the good performance of our method. Finally, applications to the Danish fire losses and to hurricane losses are presented.