Truncated Geometric Bootstrap Method for Time Series Stationary Process

ABSTRACT

This paper introduced a bootstrap method called truncated geometric bootstrap method for time series stationary process. We estimate the parameters of a geometric distribution which has been truncated as a probability model for the bootstrap algorithm. This probability model was used in resampling blocks of random length, where the length of each blocks has a truncated geometric distribution. The method was able to determine the block sizes*b* and probability *p* attached to its random selections. The mean and variance were
estimated for the truncated geometric distribution and the bootstrap algorithm
developed based on the proposed probability model.

This paper introduced a bootstrap method called truncated geometric bootstrap method for time series stationary process. We estimate the parameters of a geometric distribution which has been truncated as a probability model for the bootstrap algorithm. This probability model was used in resampling blocks of random length, where the length of each blocks has a truncated geometric distribution. The method was able to determine the block sizes

KEYWORDS

Truncated Geometric Bootstrap Method, Stationary Process, Moving Block and Geometric Stationary Bootstrap Method

Truncated Geometric Bootstrap Method, Stationary Process, Moving Block and Geometric Stationary Bootstrap Method

Cite this paper

Olatayo, T. (2014) Truncated Geometric Bootstrap Method for Time Series Stationary Process.*Applied Mathematics*, **5**, 2057-2061. doi: 10.4236/am.2014.513199.

Olatayo, T. (2014) Truncated Geometric Bootstrap Method for Time Series Stationary Process.

References

[1] Finney, D.J. (1949) The Truncated Binomial Distribution. Annals of Eugenics, 14, 319-328.

http://dx.doi.org/10.1111/j.1469-1809.1947.tb02410.x

[2] Rider, P.R. (1953) Truncated Poisson Distribution. Journal of the American Statistical Association, 48, 826-830.

http://dx.doi.org/10.1080/01621459.1953.10501204

[3] Rider, P.R. (1955) Truncated Binomial and Negative Binomial Distribution. Journal of the American Statistical Association, 50, 877-883.

http://dx.doi.org/10.1080/01621459.1955.10501973

[4] Kunsch, H.R. (1989) The Jacknife and the Bootstrap for General Stationary Observations. The Annals of Statistics, 17, 1217-1241.

http://dx.doi.org/10.1214/aos/1176347265

[5] Liu, R.Y. and Singh, K. (1992) Moving Blocks Jackknife and Bootstrap Capture Weak Dependence. In: R. Lepage and L. Billard, Eds., Exploring the Limits of Bootstrap, John Wiley, New York.

[6] Politis, D.N. and Romano, J.O. (1994) The Stationary Bootstrap. Journal of American Statistical Association, 89, 303-1313.

http://dx.doi.org/10.1080/01621459.1994.10476870

[7] Barreto, H. and Howland, F.M. (2005) Introductory Econometrics, Using Monte Carlo simulation with Microsoft Excel. Cambridge University Press, Cambridge.

http://dx.doi.org/10.1017/CBO9780511809231

[8] Efron, B. (1979) Bootstrap Methods: Another Look at the Jacknife. The Annals of Statistics, 7, 1-26.

http://dx.doi.org/10.1214/aos/1176344552

[9] Leger, C., Politis, D. and Romano, J. (1992) Bootstrap Technology and Applications. Technometrics, 34, 378-398.

http://dx.doi.org/10.1080/00401706.1992.10484950

[10] Efron, B. and Tibshirani, R. (1986) Bootstrap Measures for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy. Statistical Science, 1, 54-77.

http://dx.doi.org/10.1214/ss/1177013815

[11] Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall/CRC, London

[12] Diciccio, T. and Romano, J. (1988) A Review of Bootstrap Confidence Intervals (with Discussion). Journal of the Royal Statistical Society B, 50, 338-370.

[13] Davison, A.C. and Hinkley, D.V. (1997) Bootstrap Methods and their Application. Cambridge University Press, Cambridge.

http://dx.doi.org/10.1017/CBO9780511802843

[14] Bancroft, G.A., Colwell, D.J. and Gillet, J.R. (1983) A Truncated Poisson Distribution. The Mathematical Gazette, 66, 216-218.

[15] Kapadia, C.H. and Thomasson, R.L. (1975) On Estimating the Parameter of the Truncated Geometric Distribution by the Method of Moments. Annals of the Institute of Statistical Mathematics, 20, 519-532.

[1] Finney, D.J. (1949) The Truncated Binomial Distribution. Annals of Eugenics, 14, 319-328.

http://dx.doi.org/10.1111/j.1469-1809.1947.tb02410.x

[2] Rider, P.R. (1953) Truncated Poisson Distribution. Journal of the American Statistical Association, 48, 826-830.

http://dx.doi.org/10.1080/01621459.1953.10501204

[3] Rider, P.R. (1955) Truncated Binomial and Negative Binomial Distribution. Journal of the American Statistical Association, 50, 877-883.

http://dx.doi.org/10.1080/01621459.1955.10501973

[4] Kunsch, H.R. (1989) The Jacknife and the Bootstrap for General Stationary Observations. The Annals of Statistics, 17, 1217-1241.

http://dx.doi.org/10.1214/aos/1176347265

[5] Liu, R.Y. and Singh, K. (1992) Moving Blocks Jackknife and Bootstrap Capture Weak Dependence. In: R. Lepage and L. Billard, Eds., Exploring the Limits of Bootstrap, John Wiley, New York.

[6] Politis, D.N. and Romano, J.O. (1994) The Stationary Bootstrap. Journal of American Statistical Association, 89, 303-1313.

http://dx.doi.org/10.1080/01621459.1994.10476870

[7] Barreto, H. and Howland, F.M. (2005) Introductory Econometrics, Using Monte Carlo simulation with Microsoft Excel. Cambridge University Press, Cambridge.

http://dx.doi.org/10.1017/CBO9780511809231

[8] Efron, B. (1979) Bootstrap Methods: Another Look at the Jacknife. The Annals of Statistics, 7, 1-26.

http://dx.doi.org/10.1214/aos/1176344552

[9] Leger, C., Politis, D. and Romano, J. (1992) Bootstrap Technology and Applications. Technometrics, 34, 378-398.

http://dx.doi.org/10.1080/00401706.1992.10484950

[10] Efron, B. and Tibshirani, R. (1986) Bootstrap Measures for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy. Statistical Science, 1, 54-77.

http://dx.doi.org/10.1214/ss/1177013815

[11] Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall/CRC, London

[12] Diciccio, T. and Romano, J. (1988) A Review of Bootstrap Confidence Intervals (with Discussion). Journal of the Royal Statistical Society B, 50, 338-370.

[13] Davison, A.C. and Hinkley, D.V. (1997) Bootstrap Methods and their Application. Cambridge University Press, Cambridge.

http://dx.doi.org/10.1017/CBO9780511802843

[14] Bancroft, G.A., Colwell, D.J. and Gillet, J.R. (1983) A Truncated Poisson Distribution. The Mathematical Gazette, 66, 216-218.

[15] Kapadia, C.H. and Thomasson, R.L. (1975) On Estimating the Parameter of the Truncated Geometric Distribution by the Method of Moments. Annals of the Institute of Statistical Mathematics, 20, 519-532.