Discrimination of AR,MA and ARMA time series models

The problem of discrimination between two stationary ARMA time series models is considered, and in particular AR(p), MA(p), ARMA(1,1) models. The discriminant based on the likelihood ration leads to a quadratic form that is generally too complicated to evaluated explicitly. The discriminant can be expressed approximately as a linear combination of independent chi–squared random varianles each with one degree of freedom, the coefficients, of which are eigenvalues of cumbersome matrices. An analytical solution which gives the coefficients approximately is suggested.