Development of Cook’s Distance and DFFITs Measures Based on Robust-M Kibria–Lukman Estimator for Detecting Influential Observations
DOI:
https://doi.org/10.62054/ijdm/0303.20Abstract
Identifying influential observations is crucial in regression analysis because such points can distort parameter estimates, prediction accuracy, and overall statistical inference. Traditional diagnostics such as Cook’s Distance and DFFITs, developed under the Ordinary Least Squares (OLS) framework, perform poorly when multicollinearity and outliers coexist. This study develops two new influential diagnostic measures—Cook’s Distance in Robust-M Kibria-Lukman (Cooks_KL_M) and DFFITs in Robust-M Kibria-Lukman (DFFITs_KL_M)—derived within the Robust-M KL estimator to simultaneously accommodate outliers and multicollinearity. Approximate deletion formulas for both measures are also established using the Sherman–Morrison–Woodbury identity. The proposed diagnostics are evaluated using two real datasets characterized by severe multicollinearity and outliers. Results reveal that Cooks_KL_M performs comparably with existing diagnostics, while DFFITs_KL_M exhibits enhanced sensitivity, detecting additional influential points overlooked in previous studies. These findings demonstrate that integrating robustness with shrinkage parameters enhances diagnostic performance, offering practitioners a more reliable tool for analyzing complex datasets.
Riferimenti bibliografici
Abidoye, A. O., Ajayi, I.M. Lukman, A. F. & Ogunjobi, J. O. (2022). Unbiased Modified Two-
Parameter Estimator for the Linear Regression Model. Journal of of Scientific Research. Vol. 14(3), pp 785- 795.
Adejumo, T. J., Ayinde, K., Okegbade, A. I., Akomolafe, A. A., Oshuporu, O. A. and Koleoso,
S.O. (2024). Robust-M Kibria-Lukman Estimator for Linear Regression Model ith Outliers In the x-direction: Simulations and Applications. Science World Journal. Vol. 19(2). Doi.org/10.4314/swj.v19i2.21.
Adejumo. T. J., Amin, M., Ayinde, K. and Adewuyi, E. T. (2025). Influential diagnostics in the Linear model with Kibria-Lukman Estimator. Communications in Statistics-Simulation and Computation. Doi.org/10.1080/03610918.20252484628.
Alkasadi, N. A., Ibrahim, S., Abuzaid, A. H. M., Yusoff, M. I., Hamid, H., Waozhe, L. and
Abdrasak, A. (2019). Outlier Detection in Multiple circular Degression Model using DFFITs statistics. Sains Malaysiana 46 (7) (2019). 1557 – 1563. http://dx.doi.org/10.17576/.ism-2019-4807 – 25.
Arumairajan, S. & Kayathiri, S. (2002). A New Stochastic Restricted Two-Parameter
Estimator in Multiple Linear Regression Model. Vavuniya Journal of Science.1,38-47
Asar, Y. and Erisoglu, M. (2016). Influence diagnostics in Two-parameter Ridge
Regression. Journal of Data Science 14, (2016). 33-52. Doi: 10.6339/JDS. 201601_14(1).0003.
Ayinde, K., Lukman, A. F. and Arowolo, O. T. (2015). Robust regression diagnostics of
influential observations in linear regression model. Open Journal of Statistics. 5, 1- 11.
Bagheri, A., and Midi, H. (2009). Robust Estimations as a Remedy for multicollinearity
caused by multiple High leverage points. Journal of Mathematics and Statistics 5 (4); 311 – 321.
Bagheri, A., Midi, H. and Imon, A. H. M. R. (2010). The effect of collinearity – influential
observations on collinear Data set. A montecarlo Simulation study. Journal od Applied Sciences. 10 (18). 2086 -2093.
Belsley, D. A., Kuh, E. and Welsch, R. E. (1980). Regression Diagnostics: identifying
influence Data and sources of collinearity. Wiley and sons, New York.
http://dx.doi.org/10.1002/0471725153.
Birkes, D. and Dodge, Y. D. (1993). Alternative methods of regression, Wiley, New
York.
Cook, R. D. (1977). Detection of influential observations in linear regression.
Technometrics. 19; 15 – 18
Cook, R. D. and Weisberg, S. (1982). Residual and Influence in Regression. Chapman and Hall, New York.
Dawoud, I. and Hussein, E. (2025). Detection of Influential Observations for the regression model in the presence of multicollinearity: theory and methods. Communications in Statistics - Theory and Methods. Doi 10.1080/03610926.2024.244910.
Emami, H. and Emami, M. (2016). New influence diagnostics in ridge regression, Journal
of Applied Statistics. 43(3). 476-489. Doi: 10.1080/02664763.1070804.
Emami, H. and Emami, M. (2017). Influence diagnostics in Modified Liu –type Estimator.
Calcutta Statistical Association Bulletion 68(1&2) 82 – 91. DOI: 10.1177/0008068316668426.
Gujarati, D. N. (2003). Basic Econometrics, New Delhi; Tata McGrawHill. New York.
Hussein, E. (2021). Leverage and Influential observations on the Liu type estimator in the
linear regression model with the severe collinearity. Heliyon. Doi.org/10.1016/j.heliyon.2021.e07792.
Hussein, Y. A. and Abdalla, A. A. (2012). Generalized Two stages Ridge Regression
Estimator for Multicollinearity and Autocorrelated errors. Canadian Journal on Science and Engineering Mathematics, 3(3), 79 - 85.
Jahufer, A. (2013). Detecting Global influential observations in Liu Regression Model.
Open journal of statistics. 3, 5 – 11.http//dx.doi.org/10.4236/ojs.2013.31002.
Jahufer, A. and Jianbao, C. (2008). Assessing global influential observations in modified
ridge regression. Statistics and probability letters. 79 (2008). 513 -518. http://dx.doi.org / 10.1080/ 00401766. 1970.10488634.
Jegede, S.L., Lukman, A. F., Ayinde, K. and Odeniyi, K. A. (2022). Jacknife Kibria-Lukman M- Estimator: Simulation and Application. Journal of the Nigeria Society of Physical Science. Vol. 4, pp 251 – 264. Doi: 10.46481/jnsps. 2022.664.
Kashif, M., Amanullah, M. and Aslam, M. (2018). Pena’s statistic for the Liu regression.
Journal of Statistic for the Liu regression. Journal of Statistical Computation and Simulation 88 (13): 2473 – 2488.
Kashif, M., Ullah, M. A. and Aslam, M. (2019). Influential diagnostic with Pena’s statistic
for the Modified ridge regression. Communication in Statistics-Simulation and Computation. Doi: 10.1080/03610918.2019.1634204.
Kibria, B. M. and Lukman, A. F. (2020). A new Ridge-Type Estimator for the linear
Regression model. Simulations and Applications, Hindawi scientifica Vol. 2020. https://doi.org/10.1155/20209758378.
Kutner, M. H., Nachtsheim, C. J., Neter, J., & Li, W. (2004). Applied Linear Statistical Models (5th ed.). New York, NY: McGraw-Hill/Irwin.
Leiva, V., Barros, M. Paula, G. and Galea, M. (2006). Influence diagnostics in log –
Birnbaum banders regression models with Censored data. Journal of Computational statistics and data analysis. 51, (2007), 5694 – 5707.
Liu, K. (1993). A new class of biased estimate in linear regression. Journal of Communications
in statistics. Theory and Methods. 22:2, 393 – 402. Doi:10.1080/03610929308831027.
Longley, J. W. (1967). An appraisal of least squares programs for electronic computer
from the point of view of the use. Journal of American Statistical Association 62: 819 – 841.
Lukman, A. F.. Arowolo, O. and Ayinde, K. (2014). Some Robust Ridge Regression for
for Handling Multicollinearity and Outliers. International Journal of Sciences.
Basic and Applied Research (IJSBAR), 16(2), 192 -202.
Lukman, A. F. and Ayinde, K. (2016). Detecting observations in Two-Parameter Liu-
Ridge Estimator. Journal of Data Science. 207218 ,Doi:10.6339/JDS.201804_16(2).0001.
Lukman, A. F. and Ayinde, K. (2020). Detecting influential observations in Two-parameter Liu-
Ridge Estimator. Journal of Data Science, 16(2).0001, 201 -218. Doi:10.6339/JDS.201804.
Lukman, A. F., Ayinde, K., Okunola, A. O., Akanbi, O. B. and Onate, C. A. (2018).
Classification-Based Ridge Estimation Techniques of Alkhamisi Methods. Journal of probability and Statistical Sciences. 16(2), 2018. 165 – 181.
Majeed, A, Ahmad, S., Aslam, M. & Kashif M. (2022). A robust Kibria-Lukman estimator for linear regression model to combat multicollinearity and outliers. Concurrency and Computation: Practice and Experience. Doi.org/10.1002/cpe.7533
Meloun, M. and Militky, J. (2001). Detection of Single influential points in OLS regression
model building. Analytical Chimica Acta. Doi: 1.1016/50003.267(01)01040-6. 169 -191.
Montgomery, D. C., Peck, E. A., & Vining, G. G. (2021). Introduction to linear regression analysis (6th ed.). John Wiley & Sons
Peng, L. Y., Midi, H., Rana, S. and Fitrianto, A. (2016). Identification of multiple outliers
in a Generalized linear model with continuous variables. Journal of Mathematical
problems in Engineering . Vol. 2016. http://dx.doi.org/10.1155/2016/5840523.
Rao, C. R. (1973). Linear Statistical Inference and its applications, 22: John Wiley and
Sons.
Seber, G. A. F., & Lee, A. J. (2012). Linear regression analysis (2nd ed.). John Wiley & Sons.
Ullah, M. I., Ahmed, N., and Ahmed, S. (2013). Influential observations in Liu regression models. Pakistan Journal of Statistics, 29(3), 343–356.
Walker, E. and Birch, J. B. (1988). Influence Measures in Ridge Regression.
Technometrics, 30(2), 221 – 227.
Welsch, R. E. (1982).Influence function and regression diagnostics. Modern Data
Analysis. New york. Academic Press.
Welsch, R. E. and Kuh, E. (1977). Linear regression diagnostics. Technical Report 923-
Sloan school of management, Massachusetts Institute of Technology
Yasin, A. and Murat, E. (2016). Influence Diagnostics in Two Parameter Ridge
Regression. Journal of Data Science, 14, 33 – 52.
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