Time-reversibility, identifiability and independence of innovations for stationary time series. (English) Zbl 0753.62058
Summary: G. Weiss [J. Appl. Probab. 12, 831-836 (1975; Zbl 0322.60037)] has shown that for causal autoregressive moving-average (ARMA) models with independent and identically distributed (i.i.d.) noise, time- reversibility is essentially unique to Gaussian processes. This result extends to quite general linear processes and the extension can be used to deduce that a non-Gaussian fractionally integrated ARMA process has at most one representation as a moving average of i.i.d. random variables with finite variance.
In the proof of this uniqueness result, we use a time-reversibility argument to show that the innovations sequence ( one-step prediction residuals) of an ARMA process driven by i.i.d. non-Gaussian noise is typically not independent, a result of interest in deconvolution problems. Further, we consider the case of an ARMA process to which independent noise is added. Using a time-reversibility argument we show that the innovations of the ARMA process with added independent noise are independent if and only if both the driving noise of the process and the added noise are Gaussian.
In the proof of this uniqueness result, we use a time-reversibility argument to show that the innovations sequence ( one-step prediction residuals) of an ARMA process driven by i.i.d. non-Gaussian noise is typically not independent, a result of interest in deconvolution problems. Further, we consider the case of an ARMA process to which independent noise is added. Using a time-reversibility argument we show that the innovations of the ARMA process with added independent noise are independent if and only if both the driving noise of the process and the added noise are Gaussian.
MSC:
62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
62E10 | Characterization and structure theory of statistical distributions |
Keywords:
ARMA processes; characterizations of Gau Gaussian distribution; fractional differencing; noncausal; causal autoregressive moving-average models; Gaussian processes; general linear processes; non-Gaussian fractionally integrated ARMA process; time-reversibility; innovations sequence; one-step prediction residuals; i.i.d. non-Gaussian noise; deconvolution problems; independent noiseCitations:
Zbl 0322.60037References:
[1] | Brockwell P. J., Time Series:Theory and Methods. (1987) · Zbl 0604.62083 · doi:10.1007/978-1-4899-0004-3 |
[2] | DOI: 10.1214/aos/1176347877 · Zbl 0715.60047 · doi:10.1214/aos/1176347877 |
[3] | Donoho D., Applied Time Series Analysis 2 pp 565– (1981) |
[4] | Findley D. F., Computer Science and Statistics:The Interface pp 11– (1986) |
[5] | DOI: 10.1093/biomet/73.2.520 · Zbl 0595.62095 · doi:10.1093/biomet/73.2.520 |
[6] | DOI: 10.1093/biomet/77.1.235 · doi:10.1093/biomet/77.1.235 |
[7] | DOI: 10.1093/biomet/75.1.170 · Zbl 0644.62089 · doi:10.1093/biomet/75.1.170 |
[8] | DOI: 10.2307/2335817 · doi:10.2307/2335817 |
[9] | Kagan A. M., Characterization Problems in Mathematical Statistics. (1973) |
[10] | Lii K. S., Ann. Statist. 10 pp 1195– (1982) |
[11] | E. Masry(1990 ) Almost sure convergence analysis of autoregressive spectral estimation in additive noise. IEEE Trans. Inform. Theory.To be published. · Zbl 0709.62084 |
[12] | DOI: 10.1214/aos/1176342616 · Zbl 0317.62059 · doi:10.1214/aos/1176342616 |
[13] | DOI: 10.2307/3212945 · Zbl 0423.60043 · doi:10.2307/3212945 |
[14] | Scargle J. D., Applied Time Series Analysis 2 pp 549– (1981) |
[15] | DOI: 10.2307/3212735 · Zbl 0322.60037 · doi:10.2307/3212735 |
[16] | DOI: 10.1214/aos/1176343855 · Zbl 0378.62075 · doi:10.1214/aos/1176343855 |
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