Investigation of the Change Point in Mean of Normal Sequence Having an Outlier

Abstract

In this study, the change point in mean of the sequence of the random variables from normal distribution under the case of having an outlier in the sequence is considered. Under with this case, the maximum likelihood estimate of the change point and the estimates of the change point using robust methods are computed. The performances of the maximum likelihood method and robust methods on the estimation of the change point according to outlier locations with different sample sizes are investigated via extensive simulation studies.

Key words: Change point; outlier; maximum likelihood method; robust methods; simulation.

Keywords

• Change point; outlier; maximum likelihood method; robust methods; simulation.

References

1. Andrews, D. F., 1974. A Robust Method for Multiple Linear Regression. Technometrics, 16, 523-531.
2. Bhar, L., “Robust Regression” web page: http://www.iasri.res.in/ebook/EBADAT/3- Diagnostics%20and%20Remedial%20Measures/5- ROBUST%20REGRESSION1.pdf December 21, 2011) (accessed
3. Boudjellaba, H., MacGibbon, B. and Sawyer, P., 2001. On exact inference for change in a Poisson Sequence. Communications in Statistics A: Theory and Methods, 30(3), 407−434.
4. Chen, J. and Gupta, A. K., 1997. Testing and locating variance change points with application to stock prices. Journal of the American Statistical Association:JASA. 92(438), 739-747.
5. Chen, J. and Gupta, A. K., 2004. Statistical inference of covariance change points in Gaussian model, Statistics. 38, 17-28.
6. Fotopoulos, S. B. and Jandhyala, V. K., 2001. Maximum likelihood estimation of a change-point for exponentially distributed random variables. Statistics & Probability Letters. 51, 423-429.
7. Haccou, P., Meelis, E. and van de Geer, S., 1988. The likelihood ratio test for the change point problem for exponentially distributed random variables. Applications. 27, 121-139. and Their
8. Harvey, A. C., 1977. A comparison of preliminary estimators for robust regerssion. Journal of the American Statistical Association. 72(360), 910- 913.

9. Hinich, M, J. and Talwar, P. P., 1975. A simple method for robust regression, Journal of the American Statistical Association. 70(349), 113- 119.
10. Hinkley, D.V., 1970. Inference about the change- point in a sequence of random variables. Biometrika 57(1), 1-17.
11. Hinkley, D. V. and Hinkley, E. A., 1970. Inference about the chance-point in a sequence of binomial variables, Biometrika. 57(3), 477-488. [12] Huber, P. J., 1973. Robust regression: Asymptotics, conjectures and monte carlo, Ann. Statist. 1, 799-821.
12. Jandhyala, V. K. and Fotopoulos, S. B., 1999. Capturing the distributional behaviour of the maximum likelihood estimator of a change point. Biometrika. 86(1), 129−140.
13. Jandhyala, V. K. and Fotopoulos, S. B., 2001. Rate of convergence of the maximum likelihood estimate of a change-point, Sankhya A, 63(2), 277−285.
14. Jarrett, R. G., 1979. A note on the intervals between coal-mining disasters. Biometrika. 66(1), 191−193.
15. Pechenizkiy, M., Bakker, J., Žliobaitė, I., Ivannikov, A. and Karkkainen, T., 2009. Online Mass Flow Prediction in CFB Boilers with Explicit Detection of Sudden Concept Drift. SIGKDD Explorations. 11(2), 109-116.
16. Ramanayake, A., 2004. Tests for a change point in the shape parameter of gamma random variables. Communications in Statistics A: Theory and Methods. 33, 4, 821-833.
17. Rousseeuw, P. J. and Leroy, A. M., 1987. Robust Regression and Outlier Detection, Wiley, New York.
18. Takeuchi, J. I. and Yamanishi, K., 2006. A Unifying framework for detecting outliers and change points from time series, IEEE Transactions on Knowledge and Data Engineering. 18 (4), 482- 492.
19. O’leary, D. P., 1990. Robust regression computation using iteratively reweighted least squares, SIAM J. Matrix Anal. Appl. 11(3),.466- 480.
20. Worsley, K. J., 1986. Confidence region and test for a change-point in a sequence of exponential family random variables. Biometrika. 73(1), 91- 104.