关于球对称模型的有效预测和预测密度估计
On efficient prediction and predictive density estimation for spherically symmetric models
摘要：
假设$X,U,Y$是具有密度$$η^{d +k/2} \, f\left(η(\|x-θ|^2+ \|u\|^2 + \|y-cθ\|^2 ) \right)\,,$$的球对称分布，其中参数$θ\in \mathbb{R}^d$和$η>0$未知，并且密度$f$和常数$c >0$已知。基于观察$X=x,U=u$，我们考虑的问题，获得一个预测密度$\hat{q}(y;x,u)$ $Y$作为测量的预期Kullback-Leibler损失。一个基准程序是最小风险等变密度$\hat{q}_{mre}$，这是广义贝叶斯关于前$π(θ, η) = η^{-1}$。对于$d \geq 3$，我们得到了$\hat{q}_{mre}$的改进，并进一步表明，优势同时适用于所有$f$有限矩和有限风险条件。我们还得到，贝叶斯预测密度的谐波前$π_h(θ, η) =η^{-1} \|θ\|^{2-d}$占主导地位的$\hat{q}_{mre}$同时为所有规模的混合法线$f$。
英文摘要：
Let $X,U,Y$ be spherically symmetric distributed having density $$η^{d +k/2} \, f\left(η(\|x-θ|^2+ \|u\|^2 + \|y-cθ\|^2 ) \right)\,,$$ with unknown parameters $θ\in \mathbb{R}^d$ and $η>0$, and with known density $f$ and constant $c >0$. Based on observing $X=x,U=u$, we consider the problem of obtaining a predictive density $\hat{q}(y;x,u)$ for $Y$ as measured by the expected Kullback-Leibler loss. A benchmark procedure is the minimum risk equivariant density $\hat{q}_{mre}$, which is Generalized Bayes with respect to the prior $π(θ, η) = η^{-1}$. For $d \geq 3$, we obtain improvements on $\hat{q}_{mre}$, and further show that the dominance holds simultaneously for all $f$ subject to finite moments and finite risk conditions. We also obtain that the Bayes predictive density with respect to the harmonic prior $π_h(θ, η) =η^{-1} \|θ\|^{2-d}$ dominates $\hat{q}_{mre}$ simultaneously for all scale mixture of normals $f$.
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