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Estimation for Nonnegative First-Order Autoregressive Processes with an Unknown Location Parameter
University of Georgia, Athens, USA
University of Georgia, Athens, USA
- 1 University of Georgia, Athens, USA
- 2 University of Georgia, Athens, USA
Applied Mathematics·Volume 03 (2012)·Pages 2133–2147·Published 28 December 2012·DOI10.4236/am.2012.312A294
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Abstract
Consider a first-order autoregressive processes , where the innovations are nonnegative random variables with regular variation at both the right endpoint infinity and the unknown left endpoint θ . We propose estimates for the autocorrelation parameter f and the unknown location parameter θ by taking the ratio of two sample values chosen with respect to an extreme value criteria for f and by taking the minimum of over the observed series, where represents our estimate for f . The joint limit distribution of the proposed estimators is derived using point process techniques. A simulation study is provided to examine the small sample size behavior of these estimates.
KeywordsNonnegative Time SeriesAutoregressive ProcessesExtreme Value EstimatorRegular VariationPoint Processes
- P. J. Brockwell and R. A Davis, “Time Series: Theory and Methods,” 2nd Edition, Springer, New York, 1987.
- W. P. McCormick and G. Mathew, “Estimation for Nonnegative Autoregressive Processes with an Unknown Location Parameter,” Journal of Time Series Analysis, Vol. 14, No. 1, 1993, pp. 71-92. doi:10.1111/j.1467-9892.1993.tb00130.x
- A. E. Raftery, “Estimation Eficace Pour un Processus Autoregressif Exponentiel a Densite Discontinue,” Publications de l’Institut de Statistique de l’Université de Paris, Vol. 25, No. 1, 1980, pp. 65-91.
- R. A. Davis and W. P. McCormick, “Estimation for First-Order Autoregressive Processes with Positive or Bounded Innovations,” Stochastic Processes and Their Applications, Vol. 31, No. 1, 1989, pp. 237-250. doi:10.1016/0304-4149(89)90090-2
- A. Bartlett and W. P. McCormick, “Estimation for Non-Negative Time Series with Heavy-Tail Innovations,” Journal of Time Series Analysis, 2012. http://onlinelibrary.wiley.com/journal/10.1111/%28ISSN%291467-9892/earlyview
- S. I. Resnick, “Point Processes, Regular Variation and Weak Convergence,” Advances in Applied Probability, Vol. 18, No. 1, 1986, pp. 66-138. doi:10.2307/1427239
- Ji?í Anděl, “Non-Negative Autoregressive Processes,” Journal of Time Series Analysis, Vol. 10, No. 1, 1989, pp. 1-11. doi:10.1111/j.1467-9892.1989.tb00011.x
- Ji?í Anděl, “Non-Negative Linear Processes,” Applications of Mathematics, Vol. 36, No. 1, 1991, pp. 277-283.
- S. Datta and W. P. McCormick, “Bootstrap Inference for a First-Order Autoregression with Positive Innovations,” Journal of American Statistical Association, Vol. 90, No. 1, 1995, pp. 1289-1300. doi:10.1080/01621459.1995.10476633
- S. I. Resnick, “Heavy-Tail Phenomena: Probabilistic and Statistical Modeling,” Springer, New York, 2007.
- R. A. Davis and S. Resnick, “Limit Theory for Moving Averages of Random Variables with Regularly Varying Tail Probabilities,” Annals of Probability, Vol. 13, No. 1, 1985, pp. 179-195. doi:10.1214/aop/1176993074
- O. Kallenberg, “Random Measures,” Akademie-Verlag, Berlin, 1976.
- R. A. Davis and S. Resnick, “Limit Theory for the Sample Covariance and Correlation Functions of Moving Averages,” Annals of Statistics, Vol. 14, No. 1, 1986, pp. 533-558. doi:10.1214/aos/1176349937