Estimation with Quadratic Loss¶
James, W., & Stein, C. (1961). Estimation with Quadratic Loss. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, 361-379.
Cited by¶
2 citations across 2 artifacts.
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Domain-specific¶
- Stein's Paradox
- The dominating estimator, the James-Stein shrinkage estimator, pulls each individual estimate toward a common reference point (typically the grand mean or zero) by the factor one minus the ratio of the dimension of the problem minus two to the total squared distance of the maximum-likelihood estimate from the reference: \(\hat{\theta}^{JS} = \left(1 - \frac{p-2}{\|\bar{x}\|^2}\right)\bar{x}\) for \(p \geq 3\) parameters, where \(\bar{x}\) is the vector of sample means
This sourceDerives the James-Stein estimator with the shrinkage factor 1 - (p-2)/||x||^2 and proves it dominates the sample mean for p at least 3.
- The dominating estimator, the James-Stein shrinkage estimator, pulls each individual estimate toward a common reference point (typically the grand mean or zero) by the factor one minus the ratio of the dimension of the problem minus two to the total squared distance of the maximum-likelihood estimate from the reference: \(\hat{\theta}^{JS} = \left(1 - \frac{p-2}{\|\bar{x}\|^2}\right)\bar{x}\) for \(p \geq 3\) parameters, where \(\bar{x}\) is the vector of sample means
Mechanisms¶
- Shrinkage-Aware Expectation
- The idea has deep roots: the James-Stein result showed that shrinking individually-noisy estimates toward a common center improves them on average, the formal justification for pulling extremes back toward the mean.
This sourceShows that shrinking several noisy estimates toward a common center can reduce their aggregate squared-error risk.
- The idea has deep roots: the James-Stein result showed that shrinking individually-noisy estimates toward a common center improves them on average, the formal justification for pulling extremes back toward the mean.
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