论文标题
多元正常分布的共同约束均值协方差的MLE
MLE of Jointly Constrained Mean-Covariance of Multivariate Normal Distributions
论文作者
论文摘要
估计不受约束的平均值和协方差矩阵是统计中的一个流行话题。但是，在诸如$σμ=μ$之类的联合约束下，$ n_p（μ，σ）$的参数的估计并未受到太多关注。可以将其视为$ n（θ，θ^2）$分布中经典估计问题的多元对应物。除了在参数之间（弯曲的指数族）之间存在这种非线性约束下的通常的推断挑战之外，在估计协方差矩阵时，还必须处理对称性和积极确定性的基本要求。我们以$（μ，σ）$的约束最大似然估计量来得出非线性似然方程，并使用迭代方法来解决它们。通常，使用迭代方法计算的协方差矩阵MLE无法满足约束。我们提出了一种新型算法来修改此类（不可行的）估计量或任何其他（合理的）估计量。关键步骤是使用回归概念沿协方差矩阵的特征向量重新对准平均向量。在使用Lagrangian函数进行约束MLE时（Aitchison等，1958），Lagrange乘数与感兴趣的参数纠缠在一起，并提出了另一个计算挑战。我们通过Lagrange乘数的迭代或显式计算来处理此问题。使用随机矩阵理论的最新结果，在数据依赖性的凸集内探索了受约束MLE的位置的存在和性质。一项仿真研究说明了我们的方法论，并表明修饰的估计器的性能优于迭代方法的初始估计器。
Estimating the unconstrained mean and covariance matrix is a popular topic in statistics. However, estimation of the parameters of $N_p(μ,Σ)$ under joint constraints such as $Σμ= μ$ has not received much attention. It can be viewed as a multivariate counterpart of the classical estimation problem in the $N(θ,θ^2)$ distribution. In addition to the usual inference challenges under such non-linear constraints among the parameters (curved exponential family), one has to deal with the basic requirements of symmetry and positive definiteness when estimating a covariance matrix. We derive the non-linear likelihood equations for the constrained maximum likelihood estimator of $(μ,Σ)$ and solve them using iterative methods. Generally, the MLE of covariance matrices computed using iterative methods do not satisfy the constraints. We propose a novel algorithm to modify such (infeasible) estimators or any other (reasonable) estimator. The key step is to re-align the mean vector along the eigenvectors of the covariance matrix using the idea of regression. In using the Lagrangian function for constrained MLE (Aitchison et al. 1958), the Lagrange multiplier entangles with the parameters of interest and presents another computational challenge. We handle this by either iterative or explicit calculation of the Lagrange multiplier. The existence and nature of location of the constrained MLE are explored within a data-dependent convex set using recent results from random matrix theory. A simulation study illustrates our methodology and shows that the modified estimators perform better than the initial estimators from the iterative methods.
