On Rissanen's predictive stochastic complexity for stationary ARMA processes
The asymptotic properties of the so-called predictive stochastic complexity introduced by \textit{J. Rissanen} [Ann. Stat. 14, 1080-1100 (1986; Zbl 0602.62008); J. R. Stat. Soc., Ser. B 49, 223-239 (1987; Zbl 0654.62008)] will be considered. The importance of this concept is explained by the fact that predictive stochastic complexity was shown to be a powerful tool to solve model selection problems. We consider the almost sure asymptotic properties of the reduced predictive stochastic complexity which is purely due to parameter uncertainty. This is also sometimes called the regret. We prove that it is asymptotically equal to \(\sigma^ 2 k \log N\) almost surely, where \(\sigma^ 2\) is the variance of the input noise and \(k\) is a quasi- minimal model structure, and \(N\) is the sample size. This result complements an earlier result by \textit{L. Gerencsér} and \textit{J. Rissanen} [Proc. 25th IEEE CDC, Athens, Vol. 3, 1487-1490 (1986)] in which a lower bound for the prediction error was obtained. The almost sure asymptotic behaviour of predictive stochastic complexity has been investigated in a number of independent papers in recent years. The novelty of the present paper over competing papers is that we also present an easily computable predictive stochastic complexity based on the recursive prediction error estimator of the ARMA parameters. Moreover, our analysis easily extends to more complicated estimation and control problems.
reduced predictive stochastic complexity
scientific article; zbMATH DE number 697017
scientific article