Information tradeoffs in using the sample autocorrelation function in ARMA parameter estimation

This paper considers bounds on the statistical efficiency of estimators of the poles and zeros of an ARMA process based on estimates of the process autocorrelation function (ACF). Special attention is paid to autoregressive (AR) and AR plus white noise processes. It is seen that reducing the ARMA process data to a given set of consecutive lags of the popular lagged-product ACF estimates prior to parameter estimation increases Cramer-Rao bounds on the generalized error covariance. A parametric study of the bound deterioration for some illustrative signal and noise situations reveals some empirical strategies for choosing ACF estimate lags to preserve statistical information. Analysis is based on the relative information index (RII) [2], and derivations of the large sample Fisher's information matrix for the raw data and for the lagged-product ACF estimate of an ARMA process are included.

[1]  M. Bartlett.  On the Theoretical Specification and Sampling Properties of Autocorrelated Time‐Series , 1946 .

[2]  The asymptotic distribution of serial correlation coefficients for autoregressive processes with dependent residuals , 1954 .

[3]  Mátyás Arató,et al.  On the Sufficient Statistics for Stationary Gaussian Random Processes , 1961 .

[4]  A. M. Walker.  Large-sample estimation of parameters for autoregressive processes with moving-average residuals , 1962 .

[5]  Calyampudi Radhakrishna Rao,et al.  Linear Statistical Inference and its Applications , 1967 .

[6]  T. Hsia,et al.  On a Method for Estimating Power Spectra , 1967 .

[7]  Karl Johan Åström,et al.  On the Achievable Accuracy in Identification Problems , 1967 .

[8]  E. Hannan.  The estimation of mixed moving average autoregressive systems , 1969 .

[9]  Will Gersch,et al.  Estimation of the autoregressive parameters of a mixed autoregressive moving-average time series , 1970 .

[10]  Hirotugu Akaike,et al.  Maximum likelihood estimation of structural parameters from random vibration data , 1973 .

[11]  Daniel Graupe,et al.  Identification of autoregressive moving-average parameters of time series , 1975 .

[12]  L. Scharf,et al.  A note on covariance-invariant digital filter design and autoregressive moving average spectrum analysis , 1979 .

[13]  M. Kaveh.  High resolution spectral estimation for noisy signals , 1979 .

[14]  H. Sakai,et al.  Recursive parameter estimation of an autoregressive process disturbed by white noise , 1979 .

[15]  Mostafa Kaveh,et al.  On some suboptimum ARMA spectral estimators , 1980 .

[16]  James A. Cadzow.  ARMA Spectral estimation: A model equation error procedure , 1980, ICASSP.

[17]  A. A. Beex,et al.  Generating covariance sequences and the calculation of quantization and rounding error variances in digital filters , 1980 .

[18]  H. Tokumaru,et al.  Statistical analysis of a spectral estimator for ARMA processes , 1980 .

[19]  S. Kay.  Noise compensation for autoregressive spectral estimates , 1980 .

[20]  A. A. Beex,et al.  Covariance sequence approximation for parametric spectrum modeling , 1981 .

[21]  Yiu-Tong Chan,et al.  Spectral estimation via the high-order Yule-Walker equations , 1982 .

[22]  Benjamin Friedlander.  Recursive lattice forms for spectral estimation , 1982 .

[23]  B. Dickinson.  Structure of stationary finite observation records of discrete-time stochastic linear systems , 1982, 1982 21st IEEE Conference on Decision and Control.

[24]  J. Cadzow,et al.  Spectral estimation: An overdetermined rational model equation approach , 1982, Proceedings of the IEEE.

[25]  Benjamin Friedlander.  A recursive maximum likelihood algorithm for ARMA spectral estimation , 1982, IEEE Trans. Inf. Theory.

[26]  Mostafa Kaveh,et al.  A criterion for selecting information- preserving data reductions for use in the design of multiple-parameter estimators , 1983, IEEE Trans. Inf. Theory.

[27]  Mostafa Kaveh,et al.  Statistical efficiency of correlation-based methods for ARMA spectral estimation , 1983 .

[28]  Benjamin Friedlander,et al.  A general lower bound for parametric spectrum estimation , 1984 .