An Optimal Shrinkage Factor in Prediction of Ordered Random Effects

The problem of predicting a vector of ordered parameters or its part arises in contexts such as measurement error models, signal processing, data disclosure, and small area estimation. Often estimators of functions of the ordered random effects are obtained under strong distributional assumptions, e.g., normality. We discuss a simple generalized shrinkage estimator for predicting ordered random effects. The proposed approach is distribution free and has significant advantage when there is model misspecification. We give expression to and characterization of the optimal shrinkage parameter; the expression involves the Wasserstein distance between two modelrelated distributions. We provide a framework for estimating the distance and thereby estimating an empirical version of the oracle optimal estimator. We compare the risk for the optimal predictor to that of other distribution-free estimators. Extensive simulation results are provided to support the theoretical results.