AI 中文总结
研究真实分布为错误设定的多元t分布时估计多元正态密度的收敛性，利用渐近展开推导风险一阶和二阶项，计算多元正态和t分布下相关矩、信息矩阵及高阶累积量，包括复杂三阶和四阶累积量的推导。
AI 中文摘要
本文研究当真实分布为错误设定的多元t分布时，估计多元正态密度的收敛性。统计模型是k维正态分布族（N_k(μ,Σ)），而观测值假定服从（t_k(0,I_k,ν)），其中（ν>6）。真实分布到正态模型的信息投影首先被确定为均值为零且协方差矩阵为（ν/(ν - 2)I_k）的正态分布。主要目标是评估此信息投影与通过将最大似然估计代入模型得到的正态密度之间的期望Kullback-Leibler散度。利用估计密度的一般渐近展开，本文推导出风险的显式一阶和二阶项，作为样本量（n）、维度（k）和自由度（ν）的函数。为获得二阶项，本文计算了多元正态和多元t分布下所需的矩、信息矩阵和高阶累积量。特别是，涉及二次充分统计量的复杂三阶和四阶累积量根据其指标模式进行分类，并系统地推导其值和重数。
英文摘要
This paper investigates the convergence of an estimative multivariate normal density when the true distribution is a misspecified multivariate t-distribution. The statistical model is the family of k-dimensional normal distributions (N_k(μ,Σ)), whereas the observations are assumed to follow (t_k(0,I_k,ν)), with (ν>6). The information projection of the true distribution onto the normal model is first identified as the normal distribution with mean zero and covariance matrix (ν/(ν-2)I_k). The main objective is to evaluate the expected Kullback-Leibler divergence between this information projection and the normal density obtained by substituting the maximum likelihood estimator into the model. Using a general asymptotic expansion for estimative densities, the paper derives explicit first- and second-order terms of the risk as functions of the sample size (n), the dimension (k), and the degrees of freedom (ν). To obtain the second-order term, the paper calculates the required moments, information matrices, and higher-order cumulants under both the multivariate normal and multivariate t-distributions. In particular, the complicated third- and fourth-order cumulants involving quadratic sufficient statistics are classified according to their index patterns, and their values and multiplicities are systematically derived.
