A Generalized φ-Divergence for Asymptotically Multivariate Normal Models

I. Csiszár's ( Magyar. Tud. Akad. Mat. Kutató Int. Közl 8 (1963), 85–108) ϕ-divergence, which was considered independently by M. S. Ali and S. D. Silvey ( J. R. Statist. Soc. Ser. B 28 (1966), 131–142) gives a goodness-of-fit statistic for multinomial distributed data. We define a generalized φ-divergence that unifies the ϕ-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 χ 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.