关于加权广义逆的三个矩阵乘积的不变性

利用矩阵D-AXC的最大最小秩,给出了矩阵AXC的乘积、秩以及值域的不变性,其中X为矩阵B的各种加权广义逆.     

CO-tucker: a new method for the simultaneous analysis of a sequence of paired tables

Relationships between species and their environment are a key component to understand ecological communities. Usually, this kind of data are repeated over time or space for communities and their environment, which leads to a sequence of pairs of ecological tables, i.e. multi-way matrices. This work proposes a new method which is a combined approach of STATICO and Tucker3 techniques and deals to the problem of describing not only the stable part of the dynamics of structure–function relationships between communities and their environment (in different locations and/or at different times), but also the interactions and changes associated with the ecosystems’ dynamics. At the same time, emphasis is given to the comparison with the STATICO method on the same (real) data set, where advantages and drawbacks are explored and discussed. Thus, this study produces a general methodological framework and develops a new technique to facilitate the use of these practices by researchers. Furthermore, from this first approach with estuarine environmental data one of the major advantages of modeling ecological data sets with the CO-TUCKER model is the gain in interpretability.     

Methods to Explore Uncertainty and Bias Introduced by Job Exposure Matrices

Job exposure matrices (JEMs) are used to measure exposures based on information about particular jobs and tasks. JEMs are especially useful when individual exposure data cannot be obtained. Nonetheless, there may be other workplace exposures associated with the study disease that are not measured in available JEMs. When these exposures are also associated with the exposures measured in the JEM, biases due to uncontrolled confounding will be introduced. Furthermore, individual exposures differ from JEM measurements due to differences in job conditions and worker practices. Uncertainty may also be present at the assessor level since exposure information for each job may be imprecise or incomplete. Assigning individuals a fixed exposure determined by the JEM ignores these uncertainty sources. We examine the uncertainty displayed by bias analyses in a study of occupational electric shocks, occupational magnetic fields, and amyotrophic lateral sclerosis.     

A two-stage procedure with accelerated test for selecting the best exponential population

In this article, we discuss a two-stage procedure for selecting the largest location parameter among k(k≥2) two-parameter exponential populations(or products) from an accelerated test. The accelerated test will be conducted at a higher stress level than that of normal in the second stage. under certain assumptions between parameter and stress leveL, the two-stage selection procedure, which guarantees that the probability of correct selection is at least p^*, is proposed. At the end of the paper , we present some useful tables that serve as a guide for the needed sample size in the second stage.     

Generation of random matrices with orthonormal columns and multivariate normal wariates with given sample mean and covariance

In this paper, an algorithm for generating random matrices with orthonormal columns is introduced. As pointed out by a referee, the algorithm is almost identical to Wedderburn's (1975) unpublished method. The method can also be considered as an extension of Stewart's (1980) method, which was designed to generate random orthogonal matrices. It is found outperforming a simple extension of the QR factorization method and that of Heiberger's (1978) method. This paper also demonstrates how the algorithm can be used in generating multivariate normal variates with given sample mean and sample covariance matrix.     

关于Fan Ky不等式的推广

利用积分方法和矩阵理论研究了正定对称矩阵的行列式不等式,考虑了可逆矩阵的行列式不等式,得到了Fan Ky不等式的若干推广;应用Gram的正定性,给出了Fan Ky不等式的一个改进.     

Some results on optimal design under a first-order autoregression and on finite williams type ii designs

It was shown, essentially, by Kiefer (1961) that the type II (a) design of Williams (1952) is asymptotically universally optimum for a first-order autoregression with parameter λ >0. We investigate any optimality properties these designs have when finite. We show that small differences in the definitions of the autoregression or of the design can lead to standard results in the theory of optiaml design no longer being applicable. We include some useful results on patterned matrices.     

Ellipsoidal confidence regions for a normal covariance matrix

We obtain an asymptotic expansion of the confidence coefficient for an ellipsoidal confidence region on the elements of a normal covariance matrix. This leads to simultaneous confidence intervals on all linear functions of the elements of this matrix, which are compared with those of Roy (1954).     

On the Second‐Order Random Walk Model for Irregular Locations

Abstract. The second‐order random walk (RW2) model is commonly used for smoothing data and for modelling response functions. It is computationally efficient due to the Markov properties of the joint (intrinsic) Gaussian density. For evenly spaced locations the RW2 model is well established, whereas for irregularly spaced locations there is no well established construction in the literature. By considering the RW2 model as the solution of a stochastic differential equation (SDE), a discretely observed integrated Wiener process, it is possible to derive the density preserving the Markov properties by augmenting the state‐space with the velocities. Here, we derive a computationally more efficient RW2 model for irregular locations using a Galerkin approximation to the solution of the SDE without the need of augmenting the state‐space. Numerical comparison with the exact solution demonstrates that the error in the Galerkin approximation is small and negligible in applications.     

Wilks’ factorization of the matrix-variate Dirichlet–Riesz distributions

The Riesz distributions on positive definite symmetric matrices are used to introduce a class of Dirichlet–Riesz distributions. In addition, several distributional properties are stated. Essentially, we show the relationship between the Dirichlet–Riesz distributions of the first kind and the second kind, respectively. We derive Wilks’ factorization of the matrix-variate Dirichlet–Riesz. Further, several results on the product of Riesz and beta–Riesz matrices with a set of Dirichlet–Riesz matrices of the first kind have been derived.