Estimation of parameters in ARUMA models

LetX(n) be a time series satisfying the following ARUMA (p, d, q) models: $$U(B)A(B)X(n) = C(B)W(n)$$ whereU(B)=1+u(1)B+ ... +u(d)B ^d is a polynomial with all roots on the unit circle,A(B)=1+a(1)B+ ... +a(p)B ^p is a polynomial with all roots outside the unit circle,C(B)=1+c(1)B+...+c(q)B ^q is a polynomial which is relatively prime with the polynomialU(B)A(B),B is the backshift operator such thatBX(n)=X(n?1), and (W(n),F(n),n≥1) is a sequence of martingale differences satisfying the following conditions: $$begin{gathered} mathop {lim }limits_{n to infty } E (W(n)^2 |F(n - 1)) = sigma ^2 a.s. hfill mathop {sup}limits_{n > 1} E|W(n)|^gamma< infty for some gamma > 2. hfill end{gathered} $$ The purpose of this paper is to provide consistent estimates of the parametersp, d, q, u(j) (j=1, 2, ...d), anda(k) (k=1, 2, ...,p).