Technical Report 2023-003

Relations Between Prediction Error and Maximum Likelihood Methods in an Error-in-Variables Setting. Extended version with full proofs

Torsten Söderström

October 2023

Prediction error (PE) and maximum likelihood (ML) methods are often treated as synonyms when identifying linear dynamic systems from Gaussian data. It is shown how these methods differ when specifically dealing with errors-in-variables problems. These problems can modeled using multivariable times series with a specific internal structure. In such situations the ML estimates have lower variances than the PE estimates. Explicit expressions for the covariance matrices of the estimates are given and analyzed. For the special case when the unperturbed input is white noise it is shown that the PE estimate is not identifiable, while the ML estimates still have quite small variances. Another special case concerns non-Gaussian data. In that case a pseudo-ML estimate (using the ML criterion as if the data were Gaussian) will no longer be superior to the PE estimate in terms of error variances.

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