Statistica Sinica 6(1996), 1-22

MAXIMUM LIKELIHOOD ESTIMATION FOR NONGAUSSIAN

NONMINIMUM PHASE ARMA SEQUENCES

Keh-Shin Lii and Murray Rosenblatt

University of California, Riverside and University of California, San Diego

Abstract: We consider an approximate maximum likelihood procedure for estimating parameters of possibly noncausal and noninvertible autoregressive moving average processes driven by independent identically distributed nonGaussian noise. It is shown that the normalized approximate likelihood has a global maximum at true parameter values in the nonGaussian case. Under appropriate conditions, estimates of parameters that are solutions of likelihood equations exist, are consistent and asymptotically normal. An asymptotic covariance matrix is given. The procedure is illustrated with simulation examples of ARMA(1,1) processes.

Key words and phrases: Maximum likelihood estimates, asymptotic normality, autoregressive moving average processes, noncausal, noninvertible, nonGaussian.