A Semi-Parametric Basis for Combining Estimation Problems Under Quadratic Loss

When there is uncertainty concerning the appropriate statistical model-estimator to use in representing the data sampling process, we consider a basis for optimally combining estimation problems. The objective is to produce natural adaptive estimators that are free of subjective choices and tuning parameters. In the context of two competing multivariate linear statistical models-estimators, we demonstrate a semi-parametric Stein-like (SPSL) estimator, beta overbar (alpha hat), that under quadratic loss, has superior risk performance relative to the conventional least squares estimator. The relationship of the SPSL estimator to the family of Stein estimators is noted and asymptotic and analytic finite sample risk properties of the estimator are developed for some special cases. As an application we consider the problem of a combining two polar linear models and demonstrate a corresponding SPSL estimator. An extensive sampling experiment is used to investigate the finite sample performance of the SPSL estimator over a wide range of data sampling designs and symmetric and skewed distributions. Bootstrapping procedures are used to develop confidence sets and a basis for inference