Veres, Sándor Asymptotic distributions of likelihood ratios for overparametrized ARMA processes. (English) Zbl 0625.62074 J. Time Ser. Anal. 8, 345-357 (1987). Let \(e_ t\) be a Gaussian white noise and let \(y_ t\) be an ARMA(n,m) process generated by \[ y_ t+a_ 1y_{t-1}+...+a_ ny_{t-n}=e_ t+b_ 1e_{t-1}+...+b_ me_{t-m}. \] Assume that all roots of the polynomials \[ P(z)=1+a_ 1z+...+a_ nz^ n,\quad Q(z)=1+b_ 1z+...+b_ mz^ m \] lie in the set \(\{| z| >(1-\delta)^{- 1}\}\) where \(\delta >0\). The author considers two types of the log likelihood ratios for \(y_ t\). It is proved that their asymptotic distributions depend only on \(\delta\) and are invariant with respect to the parameters \(a_ j\), \(b_ k\). Some results of a simulation study are presented. The likelihood ratio test is proposed for model order reduction. Reviewer: J.Anděl Cited in 2 Documents MSC: 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62E20 Asymptotic distribution theory in statistics 62F05 Asymptotic properties of parametric tests Keywords:chi-square tests of fit; overparametrized ARMA processes; autoregressive moving average processes; model identification; Gaussian white noise; ARMA(n,m) process; log likelihood ratios; invariant; simulation study; likelihood ratio test; model order reduction PDFBibTeX XMLCite \textit{S. Veres}, J. Time Ser. Anal. 8, 345--357 (1987; Zbl 0625.62074) Full Text: DOI References: [1] Arato M., Linear Stochastic Systems with Constant Coefficients (Lecture Notes in Control and Information Sciences 45). (1982) · doi:10.1007/BFb0043631 [2] DOI: 10.1214/aos/1176344671 · Zbl 0406.62068 · doi:10.1214/aos/1176344671 [3] DOI: 10.2307/1425908 · Zbl 0327.62055 · doi:10.2307/1425908 [4] Gihman I. I., Introduction to the Theory of Stochastic Processes (1969) [5] DOI: 10.1093/biomet/66.1.67 · Zbl 0397.62063 · doi:10.1093/biomet/66.1.67 [6] Hall P., Martingale Limit Theory and Its Application (1980) · Zbl 0462.60045 [7] DOI: 10.1214/aos/1176345144 · Zbl 0451.62068 · doi:10.1214/aos/1176345144 [8] Hannan E. J., Essays in Statistical Science pp 403– (1982) [9] DOI: 10.1016/0005-1098(82)90003-6 · Zbl 0497.93052 · doi:10.1016/0005-1098(82)90003-6 [10] DOI: 10.1214/aos/1176345577 · Zbl 0479.62069 · doi:10.1214/aos/1176345577 [11] Priestley M. B., Spectral Analysis and Time Series 1 (1981) · Zbl 0537.62075 [12] Roussas G. G., Contiguity of Probability Measures: some Applications in Statistics (1972) · Zbl 0265.60003 · doi:10.1017/CBO9780511804373 [13] DOI: 10.2307/2335091 · doi:10.2307/2335091 [14] S. Veres (1985 ) Asymptotic distribution of a structural validation statistic. Preprint from the 7th IFAC Symp. on Identification and System Parameters and Estimations, York, England. [15] S. Veres (1985 ) On the overparametrized ML estimation of ARMA processes. Proc. 5th Pannonian Symp. on Mathematical Statistics, Visegradto be published by North Holland. · Zbl 0662.62096 [16] DOI: 10.2307/1990256 · Zbl 0063.08120 · doi:10.2307/1990256 [17] DOI: 10.1214/aoms/1177732360 · Zbl 0018.32003 · doi:10.1214/aoms/1177732360 [18] Wilks S. S., Mathematical Statistics (1962) · Zbl 0173.45805 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.