In many practical applications, prior knowledge about unknown parameters is available in the form of ordering constraints, and incorporating such information into the inferential procedure can improve the efficiency of the resulting estimators. In this paper, we consider the problem of estimating Shannon's entropy of two normal populations with ordered location parameters under a general location-invariant loss function. We derive the maximum likelihood estimator, the restricted maximum likelihood estimator, and the uniformly minimum variance unbiased estimator. Furthermore, we obtain an estimator that dominates the best affine equivariant estimator. In addition, we propose a smooth estimator that improves upon the best affine equivariant estimator. Moreover, we obtain improved estimators with respect to the Pitman closeness criterion. We consider interval estimation and construct several confidence intervals, such as asymptotic confidence intervals, bootstrap confidence intervals, highest posterior density credible intervals, and the intervals based on generalized pivotal quantities. A Monte Carlo simulation study is conducted to assess the performance of the proposed point and interval estimators. Finally, the proposed methods are illustrated by using two real data sets.
Transportable inference using target population summary statistics under covariate shift
Academy of Mathematics and Systems Science Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Department of Biostatistics Mailman School of Public Health Columbia University（哥伦比亚大学麦尔曼公共卫生学院生物统计学系）；Department of Epidemiology & Biostatistics University of California at San Francisco（旧金山大学生物统计与流行病学系）
Correlation matrix of equi-correlated normal population: fluctuation of the largest eigenvalue, scaling of the bulk eigenvalues, and stock market
给定一个尺寸为$T$的$N$维样本，并形成一个样本相关阵$\mathbf{C}$。假设$N$和$T$趋于无穷大，其中$T/N $收敛到固定的有限常数$Q>0$。如果总体是因子模型，则$\mathbf{C}$的特征值分布几乎必然弱收敛于Marčenko-Pastur分布，使得指标为PDMATHAMR，尺度参数为特定方差与第$i$个变量$(i\to\infty)$的极限比.对于具有等相关系数$ρ$的$N$维正态总体，这是一个单因子模型，对于$\mathbf{C}$的最大特征值$λ$，我们证明了$λ/N$几乎必然收敛于等相关系数$ρ$.这些结果表明了等相关正态总体和因子模型的重要作用（Laloux et al. Random matrix theory and financial correlations，Int. J. Theor. Appl. Finance，2000）：股票价格收益率样本相关阵的特征值直方图符合指数$T/N $和尺度参数$1-λ/N$的Marčenko-Pastur分布密度。此外，我们还给出了等相关正态总体样本协方差矩阵最大特征值的极限分布。我们讨论了$N$等相关系数衰减率的相变.
