Melting behavior of nylon 6 fiber in textiles

The melting process of constrained nylon 6 fibers has been studied to estimate the true melting point of its original crystals. The melting peak became simpler in shape and shifted to higher temperature with increasing fiber-axis restricting force. When heating rate, β, was increased, the temperature where the melting curve initially departs from its baseline, Tsm, decreased steeply in the range of 45 to 60°C min^-1, and increased linearly with increasing β above 60°C min^-1. By linear extrapolation of Tsm to 0°C min^-1, the temperature of ca 190°C was obtained for the melting temperature of the original nylon 6 crystals. This seems to correspond to the zero-entropy-production melting of the most imperfect crystallites of the nylon 6 fabric. This revised version was published online in July 2006 with corrections to the Cover Date.     

Estimating the covariance matrix and the generalized variance under a symmetric loss

For estimating the power of a generalized variance under a multivariate normal distribution with unknown means, the inadmissibility of the best affine equivariant estimator relative to the symmetric loss is shown, and a class of improved estimators is given. The problem of estimating the covariance matrix is also discussed.     

Estimation of Variance Components in Mixed Linear Models

In mixed linear models with two variance components, classes of estimators improving on ANOVA estimators for the variance components and the ratio of variances are constructed on the basis of the invariant statistics. Out of the classes, consistent, improved and positive estimators are singled out. These estimators are shown to be further dominated by utilizing the information contained in the noninvariant statistics. Applications to the unbalanced one-way ANOVA models and to balanced incomplete block designs are given.     

Minimax multivariate empirical Bayes estimators under multicollinearity

In this paper we consider the problem of estimating the matrix of regression coefficients in a multivariate linear regression model in which the design matrix is near singular. Under the assumption of normality, we propose empirical Bayes ridge regression estimators with three types of shrinkage functions, that is, scalar, componentwise and matricial shrinkage. These proposed estimators are proved to be uniformly better than the least squares estimator, that is, minimax in terms of risk under the Strawderman's loss function. Through simulation and empirical studies, they are also shown to be useful in the multicollinearity cases.     

Effect of dyeing on melting behavior of poly(lactic acid) fabric

The effect of the dyeing on the melting behavior of poly(lactic acid) fabrics was investigated by differential scanning calorimeter. The DSC melting peaks at 10°C min^-1 of the untreated poly(lactic acid) fabric were observed at a temperature higher than those of the dyed fabrics. The restricting force from the extended tie molecules along the fiber axis seems to decrease in the dyeing process. When the sample was rapidly heated, the crystallites melted at lower temperatures since recrystallization was restricted. It was estimated, based on the heating-rate dependency of melting behavior, that the original crystallites of the untreated sample melted at 146.1°C and those of the dyed samples melted at higher temperatures, suggesting that their crystallites are grown to be more perfect in the dyeing process. This revised version was published online in July 2006 with corrections to the Cover Date.     

Estimation of covariance matrices in fixed and mixed effects linear models

The estimation of the covariance matrix or the multivariate components of variance is considered in the multivariate linear regression models with effects being fixed or random. In this paper, we propose a new method to show that usual unbiased estimators are improved on by the truncated estimators. The method is based on the Stein–Haff identity, namely the integration by parts in the Wishart distribution, and it allows us to handle the general types of scale-equivariant estimators as well as the general fixed or mixed effects linear models.     

A unified approach to estimation of noncentrality parameters,the multiple correlation coefficient,and mixture models

We consider a class of mixture models for positive continuous data and the estimation of an underlying parameter θ of the mixing distribution. With a unified approach, we obtain classes of dominating estimators under squared error loss of an unbiased estimator, which include smooth estimators. Applications include estimating noncentrality parameters of chi-square and F-distributions, as well as ρ ²/(1 ? ρ ²), where ρ is amultivariate correlation coefficient in a multivariate normal set-up. Finally, the findings are extended to situations, where there exists a lower bound constraint on θ.     

Conditional and unconditional methods for selecting variables in linear mixed models

In the problem of selecting the explanatory variables in the linear mixed model, we address the derivation of the (unconditional or marginal) Akaike information criterion (AIC) and the conditional AIC (cAIC). The covariance matrices of the random effects and the error terms include unknown parameters like variance components, and the selection procedures proposed in the literature are limited to the cases where the parameters are known or partly unknown. In this paper, AIC and cAIC are extended to the situation where the parameters are completely unknown and they are estimated by the general consistent estimators including the maximum likelihood (ML), the restricted maximum likelihood (REML) and other unbiased estimators. We derive, related to AIC and cAIC, the marginal and the conditional prediction error criteria which select superior models in light of minimizing the prediction errors relative to quadratic loss functions. Finally, numerical performances of the proposed selection procedures are investigated through simulation studies.     

Estimating common parameters of growth curve models

Suppose that we have two independent random matrices X ₁ and X ₂ having multivariate normal distributions with common unknown matrix of parameters (q×m) and different unknown covariance matrices ₁ and ₂, given by N _{p1, N1} (B ₁ A ₁; ₁, I) and N _{p2, N2} (B ₂ A ₂; ₂, I) respectively. Let % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaigdaaaa!391C!\[\hat \xi 1\] (% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaikdaaaa!391D!\[\hat \xi 2\]) be the MLE of based on X ₁ (X ₂) only. When q=1, necessary and sufficient conditions that a combined estimator of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaigdaaaa!391C!\[\hat \xi 1\] and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaikdaaaa!391D!\[\hat \xi 2\] has uniformly smaller covariance matrix than those of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaigdaaaa!391C!\[\hat \xi 1\] and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaikdaaaa!391D!\[\hat \xi 2\] are given. The k-sample problem as well as one-sample problem is also discussed. These results are extensions of those of Graybill and Deal (1959, Biometrics, 15, 543–550), Bhattacharya (1980, Ann. Statist., 8, 205–211; 1984, Ann. Inst. Statist. Math., 36, 129–134) to multivariate case.Dedicated to Professor Yukihiro Kodama on his 60th birthday.Bowling Green State UniversityVisiting Professor on leave from the University of Tsukuba, Japan. Now at Department of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan.This research was partially supported by University of Tsukuba Project Research 1986.