An approximate maximum likelihood estimation for non-Gaussian non-minimum phase moving average processes. (English) Zbl 0765.62082

Lii, Keh-Shin; Rosenblatt, Murray

J. Multivariate Anal. 43, No. 2, 272-299 (1992).

Summary: An approximate maximum likelihood procedure is proposed for the estimation of parameters in possibly nonminimum phase (noninvertible) moving average processes driven by independent and identically distributed non-Gaussian noise. Under appropriate conditions, parameter estimates that are solutions of likelihood-like equations are consistent and are asymptotically normal. A simulation study for MA(2) processes illustrates the estimation procedure.

MSC:

62M09	Non-Markovian processes: estimation
62E20	Asymptotic distribution theory in statistics
62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)

Keywords:

spectral density; consistency; asymptotic normality; noninvertible; approximate maximum likelihood procedure; nonminimum phase; moving average processes; independent and identically distributed non-Gaussian noise; solutions of likelihood-like equations; simulation study; MA(2) processes

References:

[1]	Bartelt, H.; Lohmann, A.; Wirnitzer, B., Phase and amplitude recovery from bispectra, Appl. Opt., 23, 3121-3129 (1984)
[2]	Basor, E.; Helton, J. W., A new proof of the Szegö limit theorem and new results for Toeplitz operators with discontinuous symbol, J. Operator Theory, 3, 23-29 (1980) · Zbl 0439.47023
[3]	Böttcher, A.; Silverman, B., (Analysis of Toeplitz Operators (1990), Springer: Springer Berlin) · Zbl 0732.47029
[4]	Breidt, F. J.; Davis, R. A.; Lii, K. S.; Rosenblatt, M., Maximum likelihood estimation for noncausal autoregressive processes, J. Multivariate Anal., 36, 175-198 (1991) · Zbl 0711.62072
[5]	Donoho, D., On minimum entropy deconvolution, (Findley, D. F., Applied Time Series Analysis (1981)), 565-608 · Zbl 0481.62075
[6]	Kreiss, J., On adaptive estimation in stationary ARMA processes, Ann. Statist., 15, 112-133 (1987) · Zbl 0616.62042
[7]	Lehmann, E. L., (Theory of Point Estimation (1983), Wiley: Wiley New York) · Zbl 0522.62020
[8]	Lii, K. S.; Rosenblatt, M., Deconvolution and estimation of transfer function phase and coefficients for non-Gaussian linear processes, Ann. Statist., 10, 1195-1208 (1982) · Zbl 0512.62090
[9]	Lohmann, A.; Weigelt, G.; Wirnitzer, B., Speckle masking in astronomy: Triple correlation theory and applications, Appl. Opt., 23, 3121-3129 (1983)
[10]	Matsuoka, T.; Ulrych, T., Phase estimation using the bispectrum, (Proc. IEEE, 72 (1984)), 1403-1411
[11]	Nikias, C. L.; Raghuveer, M. R., Bispectrum estimation: A digital signal processing framework, (Proc. IEEE, 75 (1987)), 869-891
[12]	Rosenblatt, M., (Stationary Sequences and Random Fields (1985), Birkhäuser: Birkhäuser Boston) · Zbl 0597.62095
[13]	(Proceedings of the International Signal Processing Workshop on Higher Order Statistics. Proceedings of the International Signal Processing Workshop on Higher Order Statistics, Chamrousse, France (1991))
[14]	Wiggins, R. A., Minimum entropy deconvolution, Geoexploration, 17, 21-35 (1978)
[15]	Wiggins, R. A., Entropy guided deconvolution, Geophysics, 50, 2710-2716 (1985)