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Shapes of stationary autocovariances

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Robert Lund ,
Ying Zhao  and
Peter C. Kiessler
Robert Lund*
Affiliation:
Clemson University
Ying Zhao
Affiliation:
University of Georgia
Peter C. Kiessler*
Affiliation:
Clemson University
*
∗Postal address: Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975, USA.
∗Postal address: Department of Mathematical Sciences, Clemson University, Clemson, SC 29634-0975, USA.
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Abstract

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This note introduces shape orderings for stationary time series autocorrelation and partial autocorrelation functions and explores some of their convergence rate ramifications. The shapes explored include decreasing hazard rate and new better than used, orderings that are familiar from stochastic processes settings. Time series models where these shapes arise are presented. The shapes are used to obtain explicit geometric convergence rates for mean squared errors of one-step-ahead forecasts.

Keywords

ARMA processautocorrelationconvergence ratelog-convexitymean squared errorNWUpartial autocorrelation

MSC classification

Primary: 60G10: Stationary processes
Secondary: 60G20: Generalized stochastic processes
Type
Short Communications
Information
Journal of Applied Probability , Volume 43 , Issue 4 , December 2006 , pp. 1186 - 1193
Copyright
© Applied Probability Trust 2006 

Footnotes

∗∗∗

Current address: 2-2-502 Qingfeng Huajing Yuan, Haidian Qu, Beijing, 100085, P. R. China.

References

Berenhaut, K. S. and Lund, R. B. (2001). Geometric renewal convergence rates from hazard rates. J. Appl. Prob. 38, 180–194.Google Scholar
Berenhaut, K. S. and Lund, R. B. (2002). Renewal convergence rates for DHR and NWU lifetimes. Prob. Eng. Inf. Sci. 16, 67–84.Google Scholar
Billingsley, P. (1995). Probability and Measure, 3rd edn. John Wiley, New York.Google Scholar
Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd edn. Springer, New York.Google Scholar
Brown, M. (1980). Bounds, inequalities, and monotonicity properties for some specialized renewal processes. Ann. Prob. 8, 227–240.Google Scholar
Hansen, B. G. and Frenk, J. B. G. (1991). Some monotonicity properties of the delayed renewal function. J. Appl. Prob. 28, 811–821.Google Scholar
Kijima, M. (1997). Markov Processes for Stochastic Modeling. Chapman & Hall, London.CrossRefGoogle Scholar
Liggett, T. (1989). Total positivity and renewal theory. In Probability, Statistics, and Mathematics, eds Anderson, T. W., Athreya, K. B. and Iglehart, D. L., Academic Press, Boston, MA, pp. 141–162.Google Scholar
Lund, R. B., Zhao, Y. and Kiessler, P. C. (2006). A monotonicity in reversible Markov chains. J. Appl. Prob. 43, 486–499.CrossRefGoogle Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, New York.Google Scholar
Pourahmadi, M. (2001). Foundations of Time Series Analysis and Prediction Theory. John Wiley, New York.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, New York.Google Scholar