Nobuo Shinozaki

Numerical algorithms for the Moore-Penrose inverse of a matrix: Direct methods

SummaryDirect methods for computing the Moore-Penrose inverse of a matrix are surveyed, classified and tested. It is observed that the algorithms using matrix decompositions or bordered matrices are numerically more stable.

A note on estimating the common mean of k normal distributions and the stein problem

An unbiased estimator for the common mean of k normal distributions is suggested. A necessary and sufficient condition for the estimator Lo have a smaller variance than each sample mean is given. In the case of estimating the common mean vector of k p-variate (p ≤ 3) normal distributions a combined unbiased estimator may be used. We give a class of estimators which are better than the combined estimator when the loss is quadratic and the restriction of unbiasedness is removed.

Further results on the reverse-order law

An explicit expression is obtained for a pair of generalized inverses (B−,A−) such that B−A−=(AB)+MN, and a class of pairs (B−,A− of this property is shown. A necessary and sufficient condition for (AB)− to have the expression B−A− is also given.

Optimal process design in selective assembly when components with smaller variance are manufactured at three shifted means

Selective assembly is an effective approach for improving the quality of a product that is composed of two mating components, when the quality characteristic of the product is the clearance between the mating components. In this approach, the components are sorted into several groups according to their dimensions, and the product is assembled by randomly selecting mating components from corresponding groups. A number of previous studies focused on equal width partitioning schemes, in which the dimensional distributions of the two components are partitioned so that all groups have equal widths. When there is a large difference between the variances of the two component dimensions, equal width partitioning will result in a large number of surplus components due to differences between the numbers of components in corresponding groups. Some authors have proposed a method of manufacturing the component with smaller variance at three shifted means to cope with this difficulty. In the present paper, an optimal manufacturing mean design that minimises the number of surplus components is derived. It is shown that the use of the optimal design considerably reduces the number of surplus components compared with using another previously proposed manufacturing mean design and the no-shift design.

Optimal Binning Strategies under Squared Error Loss in Selective Assembly with Measurement Error

Selective assembly is an effective approach for improving a quality of a product assembled from two types of components, when the quality characteristic is the clearance between the mating components. Mease et al. (2004) have extensively studied optimal binning strategies under squared error loss in selective assembly, especially for the case when two types of component dimensions are identically distributed. However, the presence of measurement error in component dimensions has not been addressed. Here we study optimal binning strategies under squared error loss when measurement error is present. We give the equations for the optimal partition limits minimizing expected squared error loss, and show that the solution to them is unique when the component dimensions and the measurement errors are normally distributed. We then compare the expected losses of the optimal binning strategies with and without measurement error for normal distribution, and also evaluate the influence of the measurement error.

On uniqueness of two principal points for univariate location mixtures

A sufficient condition of uniqueness of two principal points is given for univariate symmetric distributions, which are not necessarily unimodal. Especially a class of location mixtures including the normal ones is shown to have unique two principal points.

Estimation of a Multivariate Normal Mean with a Class of Quadratic Loss Functions

Abstract Steins estimator for p ≧ 3 normal means is known to dominate the usual estimator if the loss is quadratic. Brown (1975) has shown that for a class of quadratic loss functions the usual estimator can be uniformly improved if the loss functions do not give such extreme weights to the coordinates that the problem essentially reduces to a two-dimensional one. However, he has not given a uniformly better estimator. In this article, we give explicit alternatives that uniformly improve on the usual estimator for a class of quadratic loss functions.

A comparison of restricted and unrestricted estimators in estimating linear functions of ordered scale parameters of two gamma distributions

The problem of estimating linear functions of ordered scale parameters of two Gamma distributions is considered. A necessary and sufficient condition on the ratio of two coefficients is given for the maximum likelihood estimator (MLE) to dominate the crude unbiased estimator (UE) in terms of mean square error. A modified MLE which satisfies the restriction is also suggested, and a necessary and sufficient condition is also given for it to dominate the admissible estimator based solely on one sample. The estimation of linear functions of variances in two sample problem and also of variance components in a one-way random effect model is mentioned.

Optimal Binning Strategies Under Squared Error Loss in Selective Assembly with a Tolerance Constraint

Selective assembly is an effective approach for improving the quality of a product assembled from two types of components when the quality characteristic is the clearance between the mating components. In this article, optimal binning strategies under squared error loss in selective assembly when the clearance is constrained by a tolerance parameter are discussed. Conditions for a set of constrained optimal partition limits are given, and uniqueness of this set is shown for the case when the dimensional distributions of the two components are identical and strongly unimodal. Some numerical results are reported that compare constrained optimal partitioning, unconstrained optimal partitioning, and equal width partitioning.

Minimaxity of empirical bayes estimators shrinking toward the grand mean when variances are unequal

Empirical Bayes estimators of a multivariate normal mean are considered when the components are independent and have unequal variances. A sufficient condition is given for the estimators to be minimax under sum. of squared error loss function when the common prior distribution is given by normal one with unkown mean and variance. A similar result is also given for the estimators which shrink toward some regression estimate.