Nonparametric empirical Bayes improvement of shrinkage estimators with applications to time series

We consider the problem of estimating a vector μ = (μ1, . . , μn) under a squared loss, based on independent observations Y_i ~ N(μ_i , 1), i = 1, . . , n, and possibly extra structural assumptions. We argue that many estimators are asymptotically equal to μ_i = αμ_i + (1 - α)Y₁+ ζ_i = μ_i + (1 - α)(Y_i- μ_i ) + ζ_i , where α ϵ [0, 1] and μ_i may depend on the data, but is not a function of Y_i, and Σ ζ ² _i = op(n). We consider the optimal estimator of the form μ_i +g(Y_i - μ_i ) for a general, possibly random, function g, and approximate it using nonparametric empirical Bayes ideas and techniques. We consider both the retrospective and the sequential estimation problems. We elaborate and demonstrate our results on the case where μ_i are Kalman filter estimators. Simulations and a real data analysis are also provided.