Maximum likelihood estimation of Wishart mean matrices under Löwner order restrictions

Abstract

For Wishart density functions, there remains a long-time question unsolved. That is whether there exists the closed-form MLEs of mean matrices over the partially Löwner ordering sets. In this note, we provide an affirmative answer by demonstrating a unified procedure on exactly how the closed-form MLEs are obtained for the simple ordering case. Under the Kullback-Leibler loss function, a property of obtained MLEs is further studied. Some applications of the obtained closed-form MLEs, including the comparison between our ML estimates and Calvin and Dykstra's [Maximum likelihood estimation of a set of covariance matrices under Löwner order restrictions with applications to balanced multivariate variance components models, Ann. Statist. 19 (1991) 850-869.] which obtained by iterative algorithm, are also made.

Modifying estimators of ordered positive parameters under the Stein loss

“…Recently, Tsai [20,21] provided the closed-forms of the REML estimators under simple order and tree order restrictions. In this section, the more general case of simultaneous estimation of certain positive-definite symmetric parameter matrices is considered.…”

“…, k, let Θ i be an unknown p × p parameter matrix of a distribution. Assume that Θ i 's are positive-definite symmetric and have order restriction in the Löwner sense (see [1,21]). The order restriction considered here follows in the definition of Calvin and Dykstra [1].…”

“…Tsai [21] derived the closed-form of the REML estimator of normal covariance matrices under the simple order restriction S M . As seen in the previous subsection, the REML estimators are better than the unbiased estimators δ UB under the weighted Stein loss (6.2) with ω i = n i for i = 1, .…”

Fast $$(1+\epsilon )$$ -Approximation of the Löwner Extremal Matrices of High-Dimensional Symmetric Matrices

“…Computing Löwner extremal matrices are useful in many applications: For example, in matrixvalued imaging [3,4] (morphological operations, filtering, denoising or image pyramid representations), in formal software verification [5], in statistical inference with domain constraints [6,7], in structure tensor of computer vision [8] (Förstner-like operators), etc.…”