James-Stein shrinkage of a vector of normal means

Positive-part James-Stein estimator \(\hat\theta_i = t + c (X_i - t)\), \(c = \max(0, 1 - k \sigma^2 / \sum_i (X_i - t)^2)\), with \(k = p - 2\) for a fixed target \(t\) and \(k = p - 3\) when the target is the grand mean estimated from the same data (Efron and Morris). Identical to the Python arm morie.fn.jamste.james_stein.

Usage

james_stein(x, target = NULL, sigma2 = 1)

Arguments

x

Numeric vector of observed means, length at least 3.

target

Fixed shrinkage target, or NULL for the grand mean.

sigma2

Known sampling variance of each coordinate.

Value

A list with value (the shrinkage factor), p, target, k, sigma2, shrinkage_factor, js_estimates, original_means and mse_reduction_bound.

References

James, W. and Stein, C. (1961). Estimation with quadratic loss. Proceedings of the Fourth Berkeley Symposium 1, 361-379.

Efron, B. and Morris, C. (1975). Data analysis using Stein's estimator and its generalizations. Journal of the American Statistical Association 70, 311-319.

Examples

james_stein(c(10, -5, 3, 0.1, -2))$value
#> [1] 0.9846823