On the Application of Bootstrap Method to Stationary Time Series Process
- 1 Mathematical Sciences Department, Olabisi Onabanjo University, Ago-Iwoye, Nigeria
Abstract
This article introduces a resampling procedure called the truncated geometric bootstrap method for stationary time series process. This procedure is based on resampling blocks of random length, where the length of each blocks has a truncated geometric distribution and capable of determining the probability p and number of block b. Special attention is given to problems with dependent data, and application with real data was carried out. Autoregressive model was fitted and the choice of order determined by Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). The normality test was carried out on the residual variance of the fitted model using Jargue-Bera statistics, and the best model was determined based on root mean square error of the forecasting values. The bootstrap method gives a better and a reliable model for predictive purposes. All the models for the different block sizes are good. They preserve and maintain stationary data structure of the process and are reliable for predictive purposes, confirming the efficiency of the proposed method.
- G. A. Bancroft, D. J. Colwell and J. R. Gillet, “A Truncated Poisson Distribution,” The Mathematical Gazette, Vol. 67, No. 441, 1983, pp 216-218. doi:10.2307/3617187
- H. Barreto and F. M. Howland, “Introductory Econometrics, Using Monte Carlo Simulation with Microsoft Excel,” Cambridge University Press, Cambridge, 2005. doi:10.1017/CBO9780511809231
- D. Brillinger, “Time Series: Data Analysis and Theory,” Holen-Day, San Francisco, 1981.
- A. C. Davison and D. V. Hinkley, “Bootstrap Methods and Their Application,” Cambridge University Press, Cambridge, 1997.
- T. Diciccio and J. Romano, “A Review of Bootstrap Confidence Intervals (with Discussion),” Journal of the Royal Statistical Society, Vol. 50, 1988, pp. 338-370.
- B. Efron, “Bootstrap Methods Another Look at the Jacknife,” Annals of Statistics, Vol. 7, No. 1, 1979, pp. 1-26. doi:10.1214/aos/1176344552
- B. Efron and R. Tibshirani, “Bootstrap Measures for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy,” Statistical Science, Vol. 1, No. 1, 1986, pp. 54-77. doi:10.1214/ss/1177013815
- B. Efron and R. Tibshirani, “An Introduction to the Bootstrap,” Chapman and Hall/CRC, London, 1993.
- D. Hinkley, “Bootstrap Method,” Journal of the Royal Statistical Society, Vol. 50, 1988, pp. 321-337.
- C. H. Kapadi and R. L. Thomasson, “On Estimating the Parameter of the Truncated Geometric Distribution by the Method of Moments,” Annals of the Institute of Statistical Mathematics, Vol. 20, 1975, pp. 519-532.
- H. R. Kunsch, “The Jacknife and the Bootstrap for General Stationary Observations,” The Annals of Statistics, Vol. 17, No. 3, 1989, pp. 1217-1241. doi:10.1214/aos/1176347265
- C. Leger, D. Politis and J. Romano, “Bootstrap Technology and Applications,” Technometrics, Vol. 34, No. 4, 1992, pp. 378-398. doi:10.1080/00401706.1992.10484950
- R. Y. Liu and K. Singh, “Moving Blocks Jackknife and Bootstrap Capture Weak Dependence,” In: R. Lepage and L. Billard, Eds., Exploring the Limits of Bootstrap, John Wiley, New York, 1992.
- J. I. Nwackukwu, “Organic Matter, the Source of Our Wealth,” An Inaugural Lecture Delivered at Oduduwa Hall, Obafemi Awolowo, University of Ile-Ife, Ile-Ife, 2007.
- J. F. Ojo, T. O. Olatayo and O. O. Alabi, “Forecasting in Subsets, Autoregressive Models and Autoprojective Modes,” Asian Journal of Scientific Research, Vol. 1, No. 5, 2008, pp. 1-11.