Estimates of Orders and Parameters in General ARMA Models

Let X(n)be a time series satisfying the following general ARMA(p,d,r,q)model: E(B)U(B)A(B)X(n)=C(B)W(n),whereC(z)is relatively prime with the polynomial E(z)U(z)A(z),B is the backshiftoperator such that BX(n)=X(n-1),and(W(n),F(n),n≥1)is a sequence ofmartingale differences. For simplicity,we shall assume throughout that the initial