Error exponents for AR order testing ## AbstractThis paper is concerned with error exponents in testing problems raised by auto-regressive (\textsc{ar}) modeling. The tests to be considered are variants of generalized likelihood ratio testing corresponding to traditional approaches to auto-regressive moving-average (ARMA) modeling estimation. In several related problems like Markov order or hidden Markov model order estimation, optimal error exponents have been determined thanks to large deviations theory. \textsc{ar} order testing is specially challenging since the natural tests rely on quadratic forms of Gaussian processes. In sharp contrast with empirical measures of Markov chains, the large deviation principles satisfied by Gaussian quadratic forms do not always admit an information-theoretical representation. Despite this impediment, we prove the existence of non-trivial error exponents for Gaussian \textsc{ar} order testing. And furthermore, we exhibit situations where the exponents are optimal. These results are obtained by showing that the log-likelihood process indexed by \textsc{ar} models of a given order satisfy a large deviation principle upper-bound with a weakened information-theoretical representation.
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