Assessing the optimum condition of multivariate second order response surface model through the asymptotic inference of the eigenvalues

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.