论文标题
具有集成$ L_1 $损失的预测密度估计器
Predictive density estimators with integrated $L_1$ loss
论文作者
论文摘要
本文解决了$ y $的密度$ q（\ | y-θ\ |^2）$ y $的有效预测密度估计的问题，该$基于$ x \ sim p（\ | | x-thime | |^2）$ for $ y，y，x，x，x，x，x，x，x，x，mathbb in \ mathbb {r}^d $。所选标准是集成的$ l_1 $损失，由$ l（θ，\ hat {q}）\，= \，\ int _ {\ mathbb {r}^d} \ big | big | hat {q}（q} {q}（y）） \，dy $和相关的常见风险，以$θ$为$θ\。对于绝对连续且严格减少$ Q $，我们确定了规模扩展改进的不可避免性$ \ hat {q} _c（y; x）\，= \，= \，\ frac {1} {c^d} q \ big（\ | y-x \ | y-x \ |^2/c^2/c^2/c^2/c^2/c^2/c^2/c^big， $ c \ in（1，c_0）$。对于$ p，q $和$ d \ geq 2 $，这一发现是普遍的，并且扩展到损失功能$γ\ big（l（θ，\ hat {q}）\ big）$，严格增加$γ$。该发现还扩展到包括更通用插件密度的比例扩展改进$ q（\ | y- \hatθ（x）\ |^2 \ big）$，当参数space $θ$是$ \ mathbb {r}^d $的紧凑子集时。介绍并评论了支配性发现的数值分析。作为补充，我们证明了$ Q $的单峰假设是对$ y |θ$的分布均匀分布在以约$θ$为中心的球上分布的案例的详细分析。在这种情况下，我们提供了一个单变量（$ d = 1 $）示例，其中最佳的equivariant估计器是插件估算器，并且在所有$ \ hat {q} _c $中，插件密度$ \ hat {q} _1 $在插件密度$ \ hat {q} _1 $中都是最佳的情况（对于$ d = 1,3 $）。
This paper addresses the problem of an efficient predictive density estimation for the density $q(\|y-θ\|^2)$ of $Y$ based on $X \sim p(\|x-θ\|^2)$ for $y, x, θ\in \mathbb{R}^d$. The chosen criteria are integrated $L_1$ loss given by $L(θ, \hat{q}) \, =\, \int_{\mathbb{R}^d} \big|\hat{q}(y)- q(\|y-θ\|^2) \big| \, dy$, and the associated frequentist risk, for $θ\in Θ$. For absolutely continuous and strictly decreasing $q$, we establish the inevitability of scale expansion improvements $\hat{q}_c(y;X)\,=\, \frac{1}{c^d} q\big(\|y-X\|^2/c^2 \big) $ over the plug-in density $\hat{q}_1$, for a subset of values $c \in (1,c_0)$. The finding is universal with respect to $p,q$, and $d \geq 2$, and extended to loss functions $γ\big(L(θ, \hat{q} ) \big)$ with strictly increasing $γ$. The finding is also extended to include scale expansion improvements of more general plug-in densities $q(\|y-\hatθ(X)\|^2 \big)$, when the parameter space $Θ$ is a compact subset of $\mathbb{R}^d$. Numerical analyses illustrative of the dominance findings are presented and commented upon. As a complement, we demonstrate that the unimodal assumption on $q$ is necessary with a detailed analysis of cases where the distribution of $Y|θ$ is uniformly distributed on a ball centered about $θ$. In such cases, we provide a univariate ($d=1$) example where the best equivariant estimator is a plug-in estimator, and we obtain cases (for $d=1,3$) where the plug-in density $\hat{q}_1$ is optimal among all $\hat{q}_c$.
