Detecting Global Influential Observations in Liu Regression Model

Aboobacker Jahufer

Department of Mathematical Sciences, Faculty of Applied Sciences, South Eastern University of Sri Lanka, Sammanthurai, Sri Lanka.

DOI: 10.4236/ojs.2013.31002                

Abstract

In linear regression analysis, detecting anomalous observations is an important step for model building process. Various influential measures based on different motivational arguments and designed to measure the influence of observations on different aspects of various regression results are elucidated and critiqued. The presence of influential observations in the data is complicated by the presence of multicollinearity. In this paper, when Liu estimator is used to mitigate the effects of multicollinearity the influence of some observations can be drastically modified. Approximate deletion formulas for the detection of influential points are proposed for Liu estimator. Two real macroeconomic data sets are used to illustrate the methodologies proposed in this paper.

Keywords

Liu Estimator; Global Influential Observations; Diagnostics; Multicollinearity; Case Deletion; Approximate Deletion Formulas

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	D. A. Belsley, E. Kuh and R. E. Welsch, “Regression Diagnostics: Identifying Influence Data and Source of Collinearity,” Wiley, New York, 1980. doi:10.1002/0471725153
[2]	E. Walker and J. B. Birch, “Influence Measures in Ridge Regression,” Technometrics, Vol. 30, No. 2, 1988, pp. 221- 227. doi:10.1080/00401706.1988.10488370
[3]	D. A. Belsley, “Conditioning Diagnostics: Collinearity and Weak Data in Regression,” Wiley, New York, 1991.
[4]	L. Shi, “Local Influence in Principal Component Analysis,” Biometrika, Vol. 84, No. 1, 1997, pp. 175-186. doi:10.1093/biomet/84.1.175
[5]	A. Jahufer and J. B. Chen, “Assessing Global Influential Observations in Modified Ridge Regression,” Statistics and Probability Letters, Vol. 79, No. 4, 2009, pp. 513- 518. doi:10.1016/j.spl.2008.09.019
[6]	A. Jahufer and J. Chen, “Measuring Local Influential Observations in Modified Ridge Regression,” Journal of Data Science, Vol. 9, No. 3, 2011, pp. 359-372.
[7]	A. Jahufer and J. B. Chen, “Identifying Local Influential Observations in Liu Estimator,” Journal of Metrika, Vol. 75, No. 3, 2012, pp. 425-438. doi:10.1007/s00184-010-0334-4
[8]	J. W. Longley, “An Appraisal of Least Squares Programs for Electronic Computer for the Point of View of the User,” Journal of American Statistical Association, Vol. 62, No. 319, 1967, pp. 819-841. doi:10.1080/01621459.1967.10500896
[9]	R. D. Cook and S. Weisberg, “Residuals and Influence in Regression,” Chapman & Hall, London, 1982.
[10]	K. Liu, “A New Class of Biased Estimate in Linear Regression,” Communications in Statistics—Theory and Methods, Vol. 22, No. 2, 1993, pp. 393-402.
[11]	A. E. Hoerl and R. W. Kennard, “Ridge Regression: Biased Estimation for Nonorthogonal Problems,” Technometrics, Vol. 12, No. 1, 1970, pp. 55-67. doi:10.1080/00401706.1970.10488634
[12]	C. Stein, “Inadmissibility of the Usual Estimator for the Mean of a Multivariate Normal Distribution,” Proceeding of the third Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, December 1954 and July-August 1955, pp. 197-206.
[13]	G. C. Mcdonald and D. I. Galarneau, “A Monte Carlo Evaluation of Some Ridge-Type Estimators,” Journal of American Statistical Association, Vol. 70, No. 350, 1975, pp. 407-416. doi:10.1080/01621459.1975.10479882
[14]	C. L. Mallows, “Some comments on Cp,” Technometrics, Vol. 15, No. 4, 1973, pp. 661-675.
[15]	G. Wahba, G. H. Golub and C. G. Heath, “Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter,” Technometrics, Vol. 24, No. 2, 1979, pp. 215-223. doi:10.1080/00401706.1979.10489751
[16]	F. Akdeniz and S. Ka?iranlar, “More on the New Biased Estimator in Linear Regression,” The Indian Journal of Statistics, Vol. 63, No. 3, 2001, pp. 321-325.
[17]	S, Ka?iranlar and S. Sakallio?in, “Combining the Liu Estimator and the Principal Component Regression Estimator,” Communications in Statistics—Theory and Methods, Vol. 30, No. 12, 2001, pp. 2699-2705.
[18]	S, Ka?iranlar, G. P. H. Styan and H. J. Werner, “A New Biased Estimator In Linear Regression and a Detailed Analysis of the Widely Analyzed Dataset on Portland Cement,” The Indian Journal of Statistics, Vol. 61, No. B3, 1999, pp. 443-459.
[19]	M. H. Hubert and P. Wijekoon, “Improvement of the Liu Estimator in Linear Regression Model,” Journal of Statistical Papers, Vol. 47, No. 3, 2006, pp. 471-479. doi:10.1007/s00362-006-0300-4
[20]	N. Torigoe and K. Ujiie, “On the Restricted Liu Estimator in the Gauss-Markov Model,” Communications in Statistics—Theory and Methods, Vol. 35, No. 9, 2006, pp. 1713-1722.
[21]	J. Mandel, “Use of the Singular Value Decomposition in Regression Analysis,” The American Statistician, Vol. 36, No. 1, 1982, pp. 15-24.
[22]	A. S. Top?uba?i and N. Billor, “A Class of Biased Estimators and Their Diagnostic Measures,” 2001. http://idari.cu.edu.tr /sempozyum/bil26.htm
[23]	H. Sun, “Macroeconomic Impact of Direct Foreign Investment in China: 1979-1996,” Blackwell Publishers Ltd., Oxford, 1998.
[24]	R. D. Cook, “Detection of Influential Observations in Linear Regression,” Technometrics, Vol. 19, No. 1, 1977, pp. 15-18. doi:10.2307/1268249
[25]	L. Shi and X. Wang, “Local Influence in Ridge Regression,” Computational Statistics & Data Analysis, Vol. 31, No. 3, 1999, pp. 341-353. doi:10.1016/S0167-9473(99)00019-5