Sains Malaysiana 50(3)(2021): 859-867

http://doi.org/10.17576/jsm-2021-5003-26

Simple and Fast Generalized - M (GM) Estimator and Its Application to Real Data Set

(Penganggar Ringkas dan Pantas Teritlak- M dan Kegunaannya ke atas Set Data Sebenar)

HABSHAH MIDI¹*, SHELAN SAIED ISMAEEL², JAYANTHI ARASAN¹ & MOHAMMED A MOHAMMED³

¹Faculty of Science and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia

²Department of Mathematics, Faculty of Science, University of Zakho, Iraq

³Al-Dewanyia Technical Institute, AUT, Iraq

Received: 1 April 2020/Accepted: 9 August 2020

ABSTRACT

It is now evident that some robust methods such as MM-estimator do not address the concept of bounded influence function, which means that their estimates still be affected by outliers in the X directions or high leverage points (HLPs), even though they have high efficiency and high breakdown point (BDP). The Generalized M(GM) estimator, such as the GM6 estimator is put forward with the main aim of making a bound for the influence of HLPs by some weight function. The limitation of GM6 is that it gives lower weight to both bad leverage points (BLPs) and good leverage points (GLPs) which make its efficiency decreases when more GLPs are present in a data set. Moreover, the GM6 takes longer computational time. In this paper, we develop a new version of GM-estimator which is based on simple and fast algorithm. The attractive feature of this method is that it only downs weights BLPs and vertical outliers (VOs) and increases its efficiency. The merit of our proposed GM estimator is studied by simulation study and well-known aircraft data set.

Keywords: DRGP; GM-estimator; high leverage points; index set equality

ABSTRAK

Beberapa kaedah teguh seperti penganggar MM telah dibuktikan tidak dapat menanangi konsep fungsi pengaruh terbatas, yang membawa maksud bahawa penganggar MM masih terjejas dengan titik terpencil dalam arah X atau dikenali sebagai titik tuasan tinggi (HLPs), walaupun ia mempunyai kecekapan dan titik musnah (BDP) yang tinggi. Penganggar -M teritlak (GM), seperti penganggar GM6 dicadangkan dengan tujuan utama membuat batasan kepada pengaruh HLPs dengan fungsi pemberat. Penganggar GM6 mempunyai kekangan dengan memberi pemberat rendah kepada GLPs, yang mengakibatkankecekapan penganggar ini menurun apabila kehadiran HLPs bertambah banyak dalam suatu set data. Tambahan pula, masa pengiraan GM6 terlalu panjang. Dalam kertas ini, kami membangunkan penganggar GM versi baru berdasarkan algoritma yang mudah dan pantas. Sifat menarik yang ada bagi kaedah ini ialah ia hanya menurunkan pemberat bagi BLPs dan VOs dengan ini kecekapannya meningkat. Merit penganggar GM yang kami cadangkan telah dikaji melalui kajian simulasi dan set data kapal terbang yang terkenal.

Kata kunci: DRGP; penganggar GM; set indek kesamaan; titik musnah tinggi

REFERENCES

Alguraibawi, M., Midi, H. & Rahmatullah Imon, A.H.M. 2015. A new robust diagnostic plot for classifying good and bad high leverage points in a multiple linear regression model. Mathematical Problems in Engineering 2015: Article ID. 279472.

Andersen, R. 2008. Modern Methods for Robust Regression - Series: Quantitatives Applications in Social Sciences. United States of America: SAGE Publications, Inc. p. 152.

Bagheri, A. & Midi, H. 2016. Diagnostic plot for the identification of high leverage collinearity-influential observations. SORT-Statistics and Operations Research Transactions 39(1): 51-70.

Chatterjee, S. & Hadi, A.S. 2006. Regression Analysis by Example. 4th ed. Hoboken, New Jersey: John Wiley & Sons, Inc. pp. 21-45.

Coakley, C.W. & Hettmansperger, T.P. 1993. A bounded influence, high breakdown, efficient regression estimator. Journal of the American Statistical Association 88(423): 872-880.

Gray, J.B. 1985. Graphics for regression diagnostics. In American Statistical Association Proceedings of the Statistical Computing Section Washington, DC: American Statistical Association. pp. 102-107.

Hekimoğlu, S. & Erenoglu, R.C. 2013. A new GM-estimate with high breakdown point. Acta Geodaetica et Geophysica 48(4): 419-437.

Hill, R.W. & Paul, W.H. 1977. Two robust alternatives to least-squares regression. Journal of the American Statistical Association 72(360a): 828-833.

Huber, P.J. 2004. Robust Statistics. Hoboken, New Jersey: John Wiler & Sons, Inc. pp. 43-72.

Leroy, A.M. & Rousseeuw, J.P. 1987. Robust Regression and Outlier Detection. Hoboken, New Jersey: John Wiler & Sons, Inc. pp. 21-74.

Lim, H.A. & Midi, H. 2016. Diagnostic robust generalized potential based on Index Set Equality (DRGP (ISE)) for the identification of high leverage points in linear model. Computational Statistics 31(3): 859-877.

Midi, H., Norazan, M.R. & Rahmatullah Imon,  A.H.M. 2009. The performance of diagnostic-robust generalized potentials for the identification of multiple high leverage points in linear regression. Journal of Applied Statistics 36(5): 507-520.

Rahmatullah Imon, A.H.M. 2005. Identifying multiple influential observations in linear regression. Journal of Applied Statistics 32(9): 929-946.

Riazoshams, H. & Midi, H. 2016. The performance of a robust multistage estimator in nonlinear regression with heteroscedastic errors. Communications in Statistics-Simulation and Computation 45(9): 3394-3415.

Rousseeuw, P.J. 1985. Multivariate estimation with high breakdown point. Mathematical Statistics and Applications 8(37): 283-297.

Rousseeuw, P.J. 1984. Least median of squares regression. Journal of the American Statistical Association 79(388): 871-880.

Rousseeuw, P.J. & Croux, C. 1993. Alternatives to the median absolute deviation. Journal of the American Statistical Association 88(424): 1273-1283.

Rousseeuw, P.J. & Van Zomeren, B.C. 1990. Unmasking multivariate outliers and leverage points. Journal of the American Statistical association 85(411): 633-639.

Salleh, R. 2013. A robust estimation method of location and scale with application in monitoring process variability. Universiti Teknologi Malaysia. Ph.D. Thesis (Unpublished).

Simpson, D.G., Ruppert, D. & Carroll, R.J. 1992.  On one-step GM estimates and stability of inferences in linear regression. Journal of the American Statistical Association 87(418): 439-450.

Stromberg, A.J., Hössjer, O. & Hawkins, D.M. 2000. The least trimmed differences regression estimator and alternatives. Journal of the American Statistical Association 95(451): 853-864.

Wilcox, R.R. 2005. Introduction to Robust Estimation and Hypothesis Testing. 2nd ed. Burlington, USA: Elsivier Inc. pp. 413-464.

Yohai, V.J. 1987. High breakdown-point and high efficiency robust estimates for regression. The Annals of Statistics 15(2): 642-656.

Yohai, V.J. & Zamar, R.H. 1988. High breakdown-point estimates of regression by means of the minimization of an efficient scale. Journal of the American Statistical Association 83(402): 406-413.

*Corresponding author; email: habshah@upm.edu.my