A Generalized [phi]-Divergence for Asymptotically Multivariate Normal Models

I. Csiszár's (Magyar. Tud. Akad. Mat. Kutató Int. Közl8 (1963), 85-108) [phi]-divergence, which was considered independently by M. S. Ali and S. D. Silvey (J. R. Statist. Soc. Ser. B28 (1966), 131-142) gives a goodness-of-fit statistic for multinomial distributed data. We define a generalized [phi]-divergence that unifies the [phi]-divergence approach with that of C. R. Rao and S. K. Mitra ("Generalized Inverse of Matrices and Its Applications," Wiley, New York, 1971) and derive weak convergence to a [chi]2 distribution under the assumption of asymptotically multivariate normal distributed data vectors. As an example we discuss the application to the frequency count in Markov chains and thereby give a goodness-of-fit test for observations from dependent processes with finite memory.