A note on minimum average risk estimators for coefficients in linear models

In this note, we have derived a set of necessary and sufficient conditions for the biased estimators analyzed by Swamy and Mehta (1976) to be better than the generalized least squares estimator of the coefficient vector in a standard linear regression model.     

Randomized response multivariate designs for categorical data

Three approaches to multivariate estimation for categorical data using randomized response (RR) are described. In the first approach, practical only for 2×2 contingency tables, a multi-proportions design is used. In the second approach, a separate RR trial is used for each variate and it is noted that the multi­variate design matrix of conditional probabilities is given by the Kroneeker product of the univariate design matrices of each trial, provided that the trials are independent of each other in a certain sense. The third approach requires only a single randomization and thus may be viewed as the use of vector response. Finally, a special-purpose bivariate design is presented.     

四元数正定阵与线性方程组Ax＝b的反问题

引入了四元数正定矩阵的概念，给出了ｎ阶四元数矩阵为正定的充要条件，得到了四元数线性方程组Ａｘ＝ｂ的反问题有正定阵解、正定自共轭阵解的充要条件及解的一般形式．     

对“关于谢邦杰的一个定理推广”一文的注记

得到了如果四元数体上的矩阵Ａ＝（ａ（ｉｊ））ｍ×ｎ的行向量都不是零向量，那么有　，这一结果改进或推广了已有文献的结论。     

Moments of generalized quadratic forms and distributions of certain multivariate test statistics

Summary Moments and distributions of quadratic forms or quadratic expressions in normal variables are available in literature. Such quadratic expressions are shown to be equivalent to a linear function of independent central or noncentral chi-square variables. Some results on linear functions of generalized quadratic forms are also available in literature. Here we consider an arbitrary linear function of matrix-variate gamma variables. Moments of the determinant of such a linear function are evaluated when the matrix-variate gammas are independently distributed. By using these results, arbitrary non-null moments as well as the non-null distribution of the likelihood ratio criterion for testing the hypothesis of equality of covariance matrices in independent multivariate normal populations are derived. As a related result, the distribution of a linear function of independent matrix-variate gamma random variables, which includes linear functions of independent Wishart matrices, is also obtained. Some properties of generalized special functions of several matrix arguments are used in deriving these results.     

关于亚正定矩阵的数值半径

给出了亚(半)正定矩阵数值半径计算的一个方法,该结果与文献[1]结果的使用范围互不包含,给出了其Gerschgorin型简单实用的下界估计式,以及其它有关结果。     

Optimal weighing designs with a string property

We consider the usual (spring balance) weighing design set-up with the design matrix having a string property meaning thereby that in every row of it, there is exactly one run of 1's (the rest of the elements being 0's). We have investigated some interesting features of such matrices and used them in deriving various optimality results.     

Testing the equality of variance-covariance matrices the robust way

The robust statistic T² _Dproposed by Tiku and Singh (1982) for testing the equality of mean vectors of two mu1 t ivariate populations is modified to test the equality of variance-covariance matrices.     

On the block lu decomposition of block toeplitz matrices using the vector yule-walker equations

It is shown that the inverse of a block Toeplitz matrix can be factored into a product of an upper block triangular, a block diagonal and a lower block triangular matrices, where the component matrices consist of the solutions of the forward and backward extended Yule-Walker equations. Also, a recursive algorithm is presented to decompose nested Toeplitz matrices, which is a generalization of the Levinson-Durbin algorithm. Its derivation is elementary. The decomposition and the algorithm are useful for solving a block Toeplitz system of simultaneous equations, particularly, which appears in the vector autoregressive moving-average model analysis.     

The effect of individual variability on the life table

A general formulation of the life table in the presence of individual jeterogeneity is presented. The possible effects of heterogeneity on the various life table functions are outlined. For the case of ordinary life tables, a method is presented for the evaluation of these effects. The proposed method id illustrated by a numerical example.