论文标题
收缩估计器主导某些熵的一些天真估计器
Shrinkage Estimators Dominating Some Naive Estimators of the Selected Entropy
论文作者
论文摘要
考虑两个以独立随机变量为特征的人群$ x_1 $和$ x_2 $，这样$ x_i，i = 1,2，$遵循伽马分布，带有未知比例参数$θ_i> 0 $，以及已知的形状参数$α> 0 $（两个群体的形状参数相同）。这里$（x_1，x_2）$可能是基于两个人群的独立随机样本的适当最小统计量。与较大（较小的）香农熵相关的人口称为“更糟糕的”（“更好”）人口。为了选择较差的（更好）人口的目标，自然选择规则是选择对应于$ \ max \ {x_1，x_2 \}〜（\ min \ {x_1，x_2 \}）$的人群的人群。该自然选择规则已知具有多种最佳特性。我们考虑在平均平方误差标准下使用自然选择规则（称为选定的熵）估算选定人群的香农熵的问题。为了改善所选熵的各种天真估计量，我们得出了一类收缩估计器，这些估计量会缩小各种天真估计器向中心熵。为此，我们首先考虑一类NAIVE估计器，其中包括线性，比例和排列模棱两可的估计器，并确定此类中的最佳估计器。美国考虑的天真估计器类别包含三个自然插件估计器。为了进一步改善最佳的天真估计量，我们考虑了一般的模棱两可估计器，并获得了主导的收缩估计量。我们还介绍了一项关于各种竞争估计器的性能的模拟研究。还报告了实际数据分析以说明提出的估计器的适用性。
Consider two populations characterized by independent random variables $X_1$ and $X_2$ such that $X_i, i=1,2,$ follows a gamma distribution with an unknown scale parameter $θ_i>0$, and known shape parameter $α>0$ (the same shape parameter for both the populations). Here $(X_1,X_2)$ may be an appropriate minimal sufficient statistic based on independent random samples from the two populations. The population associated with the larger (smaller) Shannon entropy is referred to as the "worse" ("better") population. For the goal of selecting the worse (better) population, a natural selection rule is the one that selects the population corresponding to $\max\{X_1,X_2\} ~(\min\{X_1,X_2\})$ as the worse (better) population. This natural selection rule is known to possess several optimum properties. We consider the problem of estimating the Shannon entropy of the population selected using the natural selection rule (to be referred to as the selected entropy) under the mean squared error criterion. In order to improve upon various naive estimators of the selected entropy, we derive a class of shrinkage estimators that shrink various naive estimators towards the central entropy. For this purpose, we first consider a class of naive estimators comprising linear, scale and permutation equivariant estimators and identify optimum estimators within this class. The class of naive estimators considered by us contains three natural plug-in estimators. To further improve upon the optimum naive estimators, we consider a general class of equivariant estimators and obtain dominating shrinkage estimators. We also present a simulation study on the performances of various competing estimators. A real data analysis is also reported to illustrate the applicability of proposed estimators.
