On a separation theorem for generalized eigenvalues and a problem in the analysis of sample surveys

Abstract

We obtain usable bounds for the asymptotic percentage points of chi-squared tests of fit for log-linear models fitted to contingency tables estimated from survey data, by applying some new separation inequalities for the generalized eigenvalues of a matrix X′AX with respect to a matrix X′BX, when both the matrices A and B are nonnegative definite. We also present some historical remarks on the Poincaré separation theorem for eigenvalues from which our new inequalities are shown to follow.