关于平衡损失函数的球对称分布的收缩估计
On shrinkage estimation of a spherically symmetric distribution for balanced loss functions
摘要：
考虑了基于平衡损失函数的$d$维球对称分布$X$的均值向量$θ$的估计问题：{ \bf（i）} $ωρ(\|\de-\de_{0}\|^{2}) +(1-ω)ρ(\|\de - θ\|^{2})$和{ \bf（ii）} $\ell\left(ω\|\de - \de_{0}\|^{2} +(1-ω)\|\de - θ\|^{2}\right)$，其中$δ_0$是目标估计量，$ρ$和$\ell$是递增和凹函数.对于$d\geq 4$和目标估计量$δ_0(X)=X$，我们提供了Baranchik型估计量，这些估计量支配$δ_0(X)=X$并且是极小极大的。研究结果代表了Marchand \& Strawderman（\cite{ms2020}）在两个方向上的扩展：{ \bf（a）}从法线的尺度混合到Lebesgue密度分布的球形类，{ \bf（b）}从完全单调到凹$ρ'$和$\ell'$。
英文摘要：
We consider the problem of estimating the mean vector $θ$ of a $d$-dimensional spherically symmetric distributed $X$ based on balanced loss functions of the forms: {\bf (i)} $ωρ(\|\de-\de_{0}\|^{2}) +(1-ω)ρ(\|\de - θ\|^{2})$ and {\bf (ii)} $\ell\left(ω\|\de - \de_{0}\|^{2} +(1-ω)\|\de - θ\|^{2}\right)$, where $δ_0$ is a target estimator, and where $ρ$ and $\ell$ are increasing and concave functions. For $d\geq 4$ and the target estimator $δ_0(X)=X$, we provide Baranchik-type estimators that dominate $δ_0(X)=X$ and are minimax. The findings represent extensions of those of Marchand \& Strawderman (\cite{ms2020}) in two directions: {\bf (a)} from scale mixture of normals to the spherical class of distributions with Lebesgue densities and {\bf (b)} from completely monotone to concave $ρ'$ and $\ell'$.
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