×
Lii, Keh-Shin; Rosenblatt, Murray

Bispectra and phase of non-Gaussian linear processes. (English) Zbl 0774.62089

J. Theor. Probab. 6, No. 3, 579-593 (1993).
Summary: The phase of transfer functions of linear processes which cannot be identified in the Gaussian case can be almost fully resolved in the non- Gaussian case. Estimates have been proposed in the past. A nonparametric estimate of the phase with better asymptotic convergence properties as a function of sample size is studied here. The asymptotic behavior of the bias and variance of the estimate is examined. In particular, the variance of the phase estimate is shown to be asymptotically independent of the frequency (if the frequency is not zero). Related problems are of interest in deconvolution, transfer function estimation, as well as in the resolution of astronomical images (perturbed by atmospheric turbulence) obtained by earth based telescopes.

MSC:

62M09 Non-Markovian processes: estimation
62G05 Nonparametric estimation
62G07 Density estimation

Keywords:

bispectrum; phase of transfer functions of linear processes; non-Gaussian case; asymptotic convergence properties; bias; variance of the phase estimate; deconvolution; transfer function estimation; resolution of astronomical images
Cite Review PDF
Full Text: DOI

References:

[1] Breidt, F. J., Davis, R. A., Lii, K. S. and Rosenblatt, M. (1991). Maximum likelihood estimation for noncausal autoregressive processes.J. Multiv. Anal. 36, 175-198. · Zbl 0711.62072 · doi:10.1016/0047-259X(91)90056-8
[2] Brillinger, D. R. (1977). The identification of a particular nonlinear time series system.Biometrika 64, 509-515. · Zbl 0388.62084 · doi:10.1093/biomet/64.3.509
[3] Brillinger, D., and Rosenblatt, M. (1967). Asymptotic theory of estimates ofkth order spectra. B. Harris, (ed.) J. Wiley,Spectral Analysis of Time Series, pp. 153-188.
[4] Hinich, M. (1990). Detecting a transient signal by bispectral analysis.IEEE TASSP 38, 1277-1283. · Zbl 0704.60039
[5] Kreiss, J. (1987). On adaptive estimation in stationary ARMA processes.Ann. Statist. 15, 112-133. · Zbl 0616.62042 · doi:10.1214/aos/1176350256
[6] Lii, K. S., and Rosenblatt, M. (1992). An approximate maximum likelihood estimation for non-Gaussian non-minimum phase moving average processes.J. Multivariate Analysis 43, 272-299. · Zbl 0765.62082 · doi:10.1016/0047-259X(92)90037-G
[7] Lii, K. S., and Rosenblatt, M. (1982). Deconvolution and estimation of transfer function phase and coefficients for non-Gaussian linear processes.Ann. Statist. 10, 1195-1208. · Zbl 0512.62090 · doi:10.1214/aos/1176345984
[8] Lohman, A., Weigelt, G., and Wirnitzer, B. (1983). Speckle masking in astronomy: triple correlation theory and applications.Applied Optics 22, 4028-4037. · doi:10.1364/AO.22.004028
[9] Marrow, J. C., Sanchez, P. P., and Sullivan, R. C. (1990). Unwrapping algorithm for leastsquare phase recovery from the modulo 2? bispectrum phase.J. Opt. Soc. Am. A. 7, 14-20. · doi:10.1364/JOSAA.7.000014
[10] Matsuoka, T., and Ulrych, T. (1984). Phase estimation using the bispectrum.Proc. IEEE 72, 1403-1411. · doi:10.1109/PROC.1984.13027
[11] Mendel, J. M. (1991). Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications.Proc. of IEEE 79, 278-305. · doi:10.1109/5.75086
[12] Nikias, C. C., and Raghuveer, M. R. (1987). Bispectrum estimation: a digital signal processing framework.Proc. IEEE 75, 869-891. · doi:10.1109/PROC.1987.13824
[13] Oppenheim, A. V., and Lin, J. S. (1981). The importance of phase in signals.Proc. IEEE 69, 529-541. · doi:10.1109/PROC.1981.12022
[14] Peacock, K., and Treitel, S. (1969). Predictive deconvolution: theory and practice.Geophysics 34, 155-169. · doi:10.1190/1.1440003
[15] Robinson, E. (1967). Predictive decomposition of time series with application to seismic exploration.Geophysics 32, 418-484. · doi:10.1190/1.1439873
[16] Stoer, J., and Bulirsch, R. (1980).Introduction to Numerical Analysis. Springer-Verlag, New York. · Zbl 0423.65002
[17] Tekalp, A., and Erden, A. (1989). Higher order spectrum factorization in one and two dimensions with applications in signal modeling and nonminimum phase system identification.IEEE TASSP 37, 1537-1549. · Zbl 0691.93066
[18] Wiggins, R. (1978). Minimum entropy deconvolution.Geoexploration 17, 21-35. · doi:10.1016/0016-7142(78)90005-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.