Optimclity in multirariate estimation

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Using Macintosh II with its software "Mathematics" purchased by this fiend, improved estimators for the mean matrix, genemfted variance, covariance matrix, and the latrnt roots in multivariate normal population were investigated. The sharp lower bound of the risk of the equivalent shrinkage estimator for the generalized variance is obtained. Numerical values are computed, using hypergeometric function of matrix arguments and it was seen that the risk of Stein estimator is close to the sharp lower bound when the matrix of noncennhty parameters is not so far from 0. Sequential shrinkage estimation and two-stage shrinkage estimation for the mean vector or covariance matrix are proposed, which are better than the ordinary estimators. It is shown that improved estimators for the common means or the generalized variance are obtained based on Pitman closeness criterion apart from quadratic loss or entropy loss.The locally best invariant test for outliers is derived and the numerical values of … More the power is shown, by simulation, to be larger than those of the ordinary tests . The estimation of the location parameter of the two-sided exponential distribution is considered as a nonregular distribution. It is shown that asymploticauy better estimator than the maximum likelihood estimator can be found within a class of linear combination of order statistics near the sample median, which is an extension of Akahira(1987). By simulation the values of the variance of each estimator are compared, when the sample size, is small. The Piman estimator has the smallest variance. However we were unable to obtain the asymptotic variance. The computing program of zonal polynomials made in this project will be useful for many other problems in multivariate statiidcal analysis. Graphs of the distribution of the laent roots of the bivariate Wishart matrix is obtained based on this program. Each investigator developed his study in his research theme actively and the results are reported to abroad too. Less