Excess Mean Squared Error of Empirical Bayes Estimators

Empirical Bayes (EB) estimators are widely used in statistics, machine learning, and system identification. A standard performance measure for such an estimator is the mean squared error (MSE). Unfortunately, an explicit expression for the MSE is generally unavailable for finite sample sizes$N$, and for most estimators of this type the first-order ($1/N$) asymptotic term coincides with the corresponding first-order term of$\text{MSE}(\hat{\bm \theta }^{\text{ML}})$, the MSE of the maximum likelihood estimator$\hat{\bm \theta }^{\text{ML}}$. The latter implies, for example, that standard asymptotic analysis does not reveal the performance loss incurred by hyper-parameter estimation. As a means to distinguish the performance of different estimators, we introduce the excess MSE of an EB estimator$\hat{\bm \theta }^{\text{EB}}$as the limit of$N^{2}[\text{MSE}(\hat{\bm \theta }^{\text{EB}})-\text{MSE}(\hat{\bm \theta }^{\text{ML}})]$. An explicit expression for the excess MSE of an EB estimator employing a general data-dependent hyper-parameter estimator is derived, allowing the study of the bias and variance trade-off, as well as the contribution of hyper-parameter estimation to the MSE. As specific instances, we consider kernel-based regularized estimators using the scaled EB, Stein's unbiased risk estimation, and generalized cross-validation hyper-parameter estimators. Moreover, we propose a modification to the excess MSE expression for regularized estimators, and show in simulations that this modification results in improved accuracy for moderate sample sizes.