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As documented in the literatures of response surface methodology, the optimum condition of a second order response surface model can be assessed by investigating the magnitude of the eigenvalues of the model matrix. In this paper we establish asymptotic confidence regions for the vector of eigenvalues of matrix models when the observations are obtained from a multivari- ate second order response surface model. The limiting distribution of the pivotal quantity of the vector of eigenvalues is derived by applying multidimensional delta method. Under mild condition it is shown that the sequence of the pivotal quantity of the vec- tor of eigenvalues converges to a centered multivariate normal distribution. By projection, we immediately obtain the asymptotic marginal confidence intervals for the components of each response variable. The application of the method to a real data consisting of bivariate observations is also discussed. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to establish universal kriging formula for multivariate spatial data observed over a second order stationary random field. In this study we consider multivariate isotropic parametric variogram model of power type. The parameters of the model including the cross-covariance parameter are estimated by applying least squares method. Numerical solution of the least squares equations are approximated by applying graphical approach. The main assumption adopted in this work is that the spatial process is regarded as isotropic in that the covariance function depends on the Euclidean distance among the two considered points. We demonstrate the application of the method to a bivariate spatial process of corn plants for construction the simultaneous kriging maps of the process which are generated by scatting the contour plot of the optimum prediction of the corn plants over the experimental region under bivariate parametric power type variogram model. [ABSTRACT FROM AUTHOR]

In this paper we demonstrate the application of partial sums technique in checking the validity of a multivariate linear regression model. In contrast to classical approaches where multivariate normality is always assumed, in this work we propose a test method in which the underlying distribution of the vector of response variables is unknown. In the first step the array of the observations is transformed by using multivariate set-indexed partial sums operator to the corresponding sequence of set-indexed partial sums process. The limit process is shown to be decomposable as a sum of a multivariate deterministic trend function and the p-dimensional set-indexed Brownian sheet. We show that testing the validity of the ordinary model is equivalent with that of the asymptotic model. Hence, the test problem can be handled by investigating that for the asymptotic model. Next, we define likelihood ratio test by computing the ratio between the likelihood of the trend under H0 and under H1 which are derived from the Cameron-Martin-Girsanov formula of the density function of the asymptotic model with respect to the p-dimensional Brownian sheet. The performance of the test is investigated by conducting Monte Carlo simulation. Application of the method to multivariate data is also discussed. [ABSTRACT FROM AUTHOR]