Fan and Li (2001), the SCAD estimator, with appropriate choice of the regularization (tuning) parameter, possesses a sparsity property, i.e., it estimates zero components of the true parameter vector exactly as zero with probability approaching one as sample size increases while still being consistent for the non-zero components...In other words, with appropriate choice of the regularization parameter, the asymptotic distribution of the SCAD estimator based on the overall model and that of the SCAD estimator derived from the most parsimonious correct model coincide. Fan and Li (2001) have dubbed this property the “oracle property”....It is well-known for Hodges’ estimator that the maximal (scaled) mean squared error grows without bound as sample size increases (e.g., Lehmann and Casella (1998), p.442), whereas the standard maximum likelihood estimator has constant finite quadratic risk. In this note we show that a similar unbounded risk result is in fact true for any estimator possessing the sparsity property. This means that there is a substantial price to be paid for sparsity even though the oracle property (misleadingly) seems to suggest otherwise.In "Modern statistical estimation via oracle inequalities":
Theorem 4.1. The James–Stein estimate obeys .
In other words, the James–Stein estimator is almost as good as the ideal estimator in a mean-squared error sense.The inequality (4.2) is an oracle inequality. An oracle inequality relates the performance of a real estimator with that of an ideal estimator which relies on perfect information supplied by an oracle, and which is not available in practice.
