Kreiss, Jens-Peter; Franke, Jürgen Bootstrapping stationary autoregressive moving-average models. (English) Zbl 0787.62092 J. Time Ser. Anal. 13, No. 4, 297-317 (1992). The bootstrap is applied to obtain the distribution of parameter estimators in linear time-series models. Starting with a \(\sqrt n\)- consistent estimator for the parameters of a stationary ARMA model, \(M\)- estimators are constructed. For the standardized \(M\)-estimators an asymptotic expansion is given. The leading term in this expansion is bootstrapped using a consistent estimation of the distribution of the model residuals. It is shown that this bootstrap procedure is asymptotically valid, and a simulation demonstrates that it is practically applicable. Reviewer: G.Wittwer (Dresden) Cited in 1 ReviewCited in 42 Documents MSC: 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62E20 Asymptotic distribution theory in statistics 62E15 Exact distribution theory in statistics Keywords:\(N\)-estimators; bootstrap; linear time-series models; stationary ARMA model; asymptotic expansion; consistent estimation; residuals; simulation PDFBibTeX XMLCite \textit{J.-P. Kreiss} and \textit{J. Franke}, J. Time Ser. Anal. 13, No. 4, 297--317 (1992; Zbl 0787.62092) Full Text: DOI References: [1] Ahlfors, Complex Analysis (1966) [2] Bickel, Some asymptotic theory for the bootstrap, Ann. Statist. 9 pp 1196– (1981) · Zbl 0449.62034 [3] Boldin, Estimation of the distribution of noise in an autoregressive scheme, Theor. Prob. Its Appl. 27 pp 866– (1983) [4] Bose, Edgeworth correction by bootstrap in autoregressions, Ann. Statist. 16 pp 1709– (1988) · Zbl 0653.62016 [5] Efron, Bootstrap methods:another look at the jackknife, Ann. Statist. 7 pp 1– (1979) · Zbl 0406.62024 [6] Efron, Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy, Statist. Sci. 1 pp 54– (1986) · Zbl 0587.62082 [7] Findley, Computer Science and Statistics:The Interface (1986) [8] Franke, J. and Hardle, W. (1987) On bootstrapping kernel spectral estimates. Ann. Statist., to be published. [9] Freedman, Bootstrapping regression models, Ann. Statist. 9 pp 1218– (1981) · Zbl 0449.62046 [10] Freedman, On bootstrapping two-stage least-squares estimates in stationary linear models, Ann. Statist. 12 pp 827– (1984) · Zbl 0542.62051 [11] Freedman, Bootstrapping an economic model:some empirical results, J. Bus. Econ. Statist. 2 pp 150– (1984) [12] Kreiss, A note on M estimation in stationary ARMA processes., Statist. Decis. 3 pp 317– (1985) [13] Kreiss, On adaptive estimation in stationary ARMA processes, Ann. Statist. 15 pp 112– (1987) · Zbl 0616.62042 [14] Kreiss, J.-P. (1990) Estimation of the distribution function of noise in stationary processes. Metrika. to be published. · Zbl 0735.62085 [15] Lee, C. H. and Martin, R. D. (1985) M-estimates for ARMA processes. Preprint, University of Washington. Seattle. [16] Lohse, Consistency of the bootstrap, Statist. Decis. 5 pp 353– (1987) · Zbl 0626.62038 [17] Paparoditis, E. and Streitberg, B. (1990) Order identification statistics in stationary ARMA models:vector autocorrelations and the bootstrap. Technical Report, Economics Department. Free University, Berlin. · Zbl 0752.62066 [18] Ramos, E. (1988) Resampling methods for time series. Ph.D. Thesis, Harvard University. Cambridge. MA. [19] Swanepoel, The bootstrap applied to power spectral density function estimation, Biometrika 73 pp 135– (1986) 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.