A James-Stein Type Estimator for Combining Unbiased and Possibly Biased Estimators

We present a method for combining unbiased sample data with possibly biased auxiliary information. The estimator we derive is similar in spirit to the James–Stein estimator. We prove that the estimator dominates the sample mean under quadratic loss. When the auxiliary information is unbiased, our estimator has risk slightly greater than the usual combined estimator. As the bias increases, however, the risk of the usual estimator is unbounded, while the risk of our estimator is bounded by the risk of the sample mean. We show how our estimator can be considered an approximation to the best linear combination of the sample data and the auxiliary information, allude to how it can be derived as an empirical Bayes estimator, and suggest a method for constructing confidence sets. Finally, the performance of our estimator is compared to that of the sample mean and the usual combined estimator using real forestry data.