AM  Vol.3 No.12 A , December 2012
Estimation for Nonnegative First-Order Autoregressive Processes with an Unknown Location Parameter
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.
Cite this paper: A. Bartlett and W. McCormick, "Estimation for Nonnegative First-Order Autoregressive Processes with an Unknown Location Parameter," Applied Mathematics, Vol. 3 No. 12, 2012, pp. 2133-2147. doi: 10.4236/am.2012.312A294.

[1]   P. J. Brockwell and R. A Davis, “Time Series: Theory and Methods,” 2nd Edition, Springer, New York, 1987.

[2]   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

[3]   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.

[4]   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

[5]   A. Bartlett and W. P. McCormick, “Estimation for Non-Negative Time Series with Heavy-Tail Innovations,” Journal of Time Series Analysis, 2012.

[6]   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

[7]   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

[8]   Ji?í Anděl, “Non-Negative Linear Processes,” Applications of Mathematics, Vol. 36, No. 1, 1991, pp. 277-283.

[9]   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

[10]   S. I. Resnick, “Heavy-Tail Phenomena: Probabilistic and Statistical Modeling,” Springer, New York, 2007.

[11]   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

[12]   O. Kallenberg, “Random Measures,” Akademie-Verlag, Berlin, 1976.

[13]   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