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Linear prediction of ARMA processes with infinite variance


  • Cline, Daren B. H.
  • Brockwell, Peter J.


In order to predict unobserved values of a linear process with infinite variance, we introduce a linear predictor which minimizes the dispersion (suitably defined) of the error distribution. When the linear process is driven by symmetric stable white noise this predictor minimizes the scale parameter of the error distribution. In the more general case when the driving white noise process has regularly varying tails with index [alpha], the predictor minimizes the size of the error tail probabilities. The procedure can be interpreted also as minimizing an appropriately defined l[alpha]-distance between the predictor and the random variable to be predicted. We derive explicitly the best linear predictor of Xn+1 in terms of X1,..., Xn for the process ARMA(1, 1) and for the process AR(p). For higher order processes general analytic expressions are cumbersome, but we indicate how predictors can be determined numerically.

Suggested Citation

  • Cline, Daren B. H. & Brockwell, Peter J., 1985. "Linear prediction of ARMA processes with infinite variance," Stochastic Processes and their Applications, Elsevier, vol. 19(2), pages 281-296, April.
  • Handle: RePEc:eee:spapps:v:19:y:1985:i:2:p:281-296

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    Cited by:

    1. Hill, Jonathan B. & Aguilar, Mike, 2013. "Moment condition tests for heavy tailed time series," Journal of Econometrics, Elsevier, vol. 172(2), pages 255-274.
    2. Balakrishna, N. & Hareesh, G., 2009. "Statistical signal extraction using stable processes," Statistics & Probability Letters, Elsevier, vol. 79(7), pages 851-856, April.
    3. Piotr Kokoszka & Michael Wolf, 2002. "Subsampling the mean of heavy-tailed dependent observations," Economics Working Papers 600, Department of Economics and Business, Universitat Pompeu Fabra.
    4. Bhansali, R. J. & Kokoszka, P. S., 2002. "Computation of the forecast coefficients for multistep prediction of long-range dependent time series," International Journal of Forecasting, Elsevier, vol. 18(2), pages 181-206.
    5. Karcher, Wolfgang & Shmileva, Elena & Spodarev, Evgeny, 2013. "Extrapolation of stable random fields," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 516-536.
    6. Mohammadi, Mohammad & Mohammadpour, Adel, 2009. "Best linear prediction for [alpha]-stable random processes," Statistics & Probability Letters, Elsevier, vol. 79(21), pages 2266-2272, November.
    7. Kokoszka, Piotr S. & Taqqu, Murad S., 1995. "Fractional ARIMA with stable innovations," Stochastic Processes and their Applications, Elsevier, vol. 60(1), pages 19-47, November.


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