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Asymptotic properties of risks ratios of shrinkage estimators. (English) Zbl 1348.62073
Summary: We study the estimation of the mean \(\theta\) of a multivariate normal distribution \(N_p(\theta,\sigma^2 I_p)\) in \(\mathbb{R}^p\), \(\sigma^2\) is unknown and estimated by the chi-square variable \(S^2\sim\sigma^2 \chi^2_n\).
In this work we are interested in studying bounds and limits of risk ratios of shrinkage estimators to the maximum likelihood estimator, when \(n\) and \(p\) tend to infinity provided that \(\lim_{p\to\infty} {\|\theta\|^2\over p\sigma^2}= c\). The risk ratio for this class of estimators has a lower bound \(B_m= {c\over 1+c}\), when \(n\) and \(p\) tend to infinity provided that \(\lim_{p\to\infty} {\|\theta\|^2\over p\sigma^2}= c\). We give simple conditions for shrinkage minimax estimators, to attain the limiting lower bound \(B_m\).
We also show that the risk ratio of James-Stein estimator and those that dominate it, attain this lower bound \(B_m\) (in particularly its positive-part version). We graph the corresponding risk ratios for estimators of James-Stein \(\delta_{JS}\), its positive part \(\delta^+_{JS}\), that of a minimax estimator, and an estimator dominating the James-Stein estimator in the sense of the quadratic risk (polynomial estimators proposed by T. F. Li and W. H. Kuo [Commun. Stat., Theory Methods 11, 2249–2257 (1982; Zbl 0501.62065)]) for some values of \(n\) and \(p\).

MSC:
62F12 Asymptotic properties of parametric estimators
62C20 Minimax procedures in statistical decision theory
62J07 Ridge regression; shrinkage estimators (Lasso)
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