Hisayuki Tsukuma

Generalized Bayes minimax estimation of the normal mean matrix with unknown covariance matrix

This paper addresses the problem of estimating the normal mean matrix in the case of unknown covariance matrix. This problem is solved by considering generalized Bayesian hierarchical models. The resulting generalized Bayes estimators with respect to an invariant quadratic loss function are shown to be matricial shrinkage equivariant estimators and the conditions for their minimaxity are given.

Estimation of a high-dimensional covariance matrix with the Stein loss

The problem of estimating a normal covariance matrix is considered from a decision-theoretic point of view, where the dimension of the covariance matrix is larger than the sample size. This paper addresses not only the nonsingular case but also the singular case in terms of the covariance matrix. Based on James and Steins minimax estimator and on an orthogonally invariant estimator, some classes of estimators are unifiedly defined for any possible ordering on the dimension, the sample size and the rank of the covariance matrix. Unified dominance results on such classes are provided under a Stein-type entropy loss. The unified dominance results are applied to improving on an empirical Bayes estimator of a high-dimensional covariance matrix.

A unified approach to estimating a normal mean matrix in high and low dimensions

This paper addresses the problem of estimating the normal mean matrix with an unknown covariance matrix. Motivated by an empirical Bayes method, we suggest a unified form of the Efron-Morris type estimators based on the Moore-Penrose inverse. This form not only can be defined for any dimension and any sample size, but also can contain the Efron-Morris type or Baranchik type estimators suggested so far in the literature. Also, the unified form suggests a general class of shrinkage estimators. For shrinkage estimators within the general class, a unified expression of unbiased estimators of the risk functions is derived regardless of the dimension of covariance matrix and the size of the mean matrix. An analytical dominance result is provided for a positive-part rule of the shrinkage estimators.

Minimax covariance estimation using commutator subgroup of lower triangular matrices

This paper deals with the problem of estimating the normal covariance matrix relative to the Stein loss. The main interest concerns a new class of estimators which are invariant under a commutator subgroup of lower triangular matrices. The minimaxity of a James-Stein type invariant estimator under the subgroup is shown by means of a least favorable sequence of prior distributions. The class yields improved estimators on the James-Stein type invariant and minimax estimator.

Modifying estimators of ordered positive parameters under the Stein loss

This paper treats the problem of estimating positive parameters restricted to a polyhedral convex cone which includes typical order restrictions, such as simple order, tree order and umbrella order restrictions. In this paper, two methods are used to show the improvement of order-preserving estimators over crude non-order-preserving estimators without any assumption on underlying distributions. One is to use Fenchels duality theorem, and then the superiority of the isotonic regression estimator is established under the general restriction to polyhedral convex cones. The use of the Abel identity is the other method, and we can derive a class of improved estimators which includes order-statistics-based estimators in the typical order restrictions. When the underlying distributions are scale families, the unbiased estimators and their order-restricted estimators are shown to be minimax. The minimaxity of the restrictedly generalized Bayes estimator against the prior over the restricted space is also demonstrated in the two dimensional case. Finally, some examples and multivariate extensions are given.

Shrinkage priors for Bayesian estimation of the mean matrix in an elliptically contoured distribution

This paper deals with the problem of estimating the mean matrix in an elliptically contoured distribution with unknown scale matrix. The Laplace and inverse Laplace transforms of the density allow us not only to evaluate the risk function with respect to a quadratic loss but also to simplify expressions of Bayes estimators. Consequently, it is shown that generalized Bayes estimators against shrinkage priors dominate the unbiased estimator.

Estimation of the Mean Vector in a Singular Multivariate Normal Distribution

This paper addresses the problem of estimating the mean vector of a singular multivariate normal distribution with an unknown singular covariance matrix. The maximum likelihood estimator is shown to be minimax relative to a quadratic loss weighted by the Moore–Penrose inverse of the covariance matrix. An unbiased risk estimator relative to the weighted quadratic loss is provided for a Baranchik type class of shrinkage estimators. Based on the unbiased risk estimator, a sufficient condition for the minimaxity is expressed not only as a differential inequality, but also as an integral inequality. Also, generalized Bayes minimax estimators are established by using an interesting structure of singular multivariate normal distribution.

Proper Bayes and minimax predictive densities related to estimation of a normal mean matrix

This paper deals with the problem of estimating predictive densities of a matrix-variate normal distribution with known covariance matrix. Our main aim is to establish some Bayesian predictive densities related to matricial shrinkage estimators of the normal mean matrix. The Kullback–Leibler loss is used for evaluating decision-theoretic optimality of predictive densities. It is shown that a proper hierarchical prior yields an admissible and minimax predictive density. Also, some minimax predictive densities are derived from superharmonicity of prior densities.

Implication of overexpression of dishevelled-associated activator of morphogenesis 1 (Daam-1) for the pathogenesis of human Idiopathic Pulmonary Arterial Hypertension (IPAH)

BackgroundIdiopathic pulmonary arterial hypertension (IPAH) is a rare, fatal disease of unknown pathogenesis. Evidence from our recent study suggests that IPAH pathogenesis is related to upregulation of the Wnt/planar cell polarity (Wnt/PCP) pathway. We used microscopic observation and immunohistochemical techniques to identify expression patterns of cascading proteins—namely Wnt-11, dishevelled-2 (Dvl-2), and dishevelled-associated activator of morphogenesis 1 (Daam-1)—in pulmonary arteries.MethodsWe analyzed sections of formalin-fixed and paraffin-embedded autopsied lung tissues obtained from 9 IPAH cases, 7 associated pulmonary arterial hypertension cases, and 16 age-matched controls without pulmonary arterial abnormalities. Results of microscopic observation were analyzed in relation to the cellular components and size of pulmonary arteries.ResultsVarying rates of positive reactivity to Dvl-2 and Daam-1 were confirmed in all cellular components of pulmonary arteries, namely, endothelial cells, myofibroblasts, and medial smooth muscle cells. In contrast, none of these components was reactive to Wnt-11. No specific expression patterns were observed for endothelial cells or myofibroblasts under any experimental conditions. However, marked expression of Dvl-2 and Daam-1 was confirmed in smooth muscle cells. In addition, Dvl-2 was depleted while Daam-1 expression was elevated in IPAH, in contrast with specimens from associated pulmonary arterial hypertension cases and controls.ConclusionsHigh Daam-1 expression may upregulate the Wnt/PCP pathway and cause IPAH.

Minimax estimation of a normal covariance matrix with the partial Iwasawa decomposition

This paper addresses the problem of estimating the normal covariance matrix relative to the Stein loss. The partial Iwasawa decomposition is used to reduce the original estimation problem to simultaneous estimation for variances and means of some normal distributions. The variances and the means are closely related to, respectively, the diagonal and the below-diagonal elements of a lower triangular matrix which is made from the Cholesky decomposition of the covariance matrix. Shrinkage type procedures are proposed for improvements not only on the diagonal elements but also on the below-diagonal elements corresponding to the James and Stein minimax estimator of the covariance matrix.