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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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