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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 Cramér-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) , 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.