Generalized Cook’s Distance and DFFITs for the New Biased-Based Estimator in the Presence of Multicollinearity and Outliers
DOI:
https://doi.org/10.62054/ijdm/0303.210Abstract
Regression analysis is a powerful tool for modeling relationships between variables, but its reliability hinges on meeting key assumptions of the classical linear regression model. When multicollinearity and outliers are simultaneously present, traditional estimation techniques such as Ordinary Least Squares (OLS) become unreliable, and classical influence diagnostics like Cook’s Distance and DFFITs may fail to detect influential observations accurately. To address this gap, this study develops generalized versions of Cook’s Distance and DFFITs tailored to the New Biased-Based (NBB) estimator a two-parameter estimator designed to mitigate multicollinearity. Using case-deletion and approximate analytical approaches, we derive influence diagnostics that incorporate the structure of the NBB estimator. The proposed methods are then applied to economic and manufacturing datasets known to exhibit both multicollinearity and outliers. Results demonstrate that the new diagnostic measures, viz; Cook’s D in NBB (CD_NBB) and DFFITs in NBB (DF_NBB) are effective in identifying influential observations, showing strong alignment with and, in some cases, improvements over existing robust and biased diagnostic techniques. These findings suggest that the NBB-based influence diagnostics offer a valuable addition to the toolkit for regression analysis under complex data conditions.
References
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, A., Asar, Y., and Yildiz, O. (2019). Influence diagnostics in multiple circular regression models. Communications in Statistics - Simulation and Computation, 48(9), 2786–2799. https://doi.org/10.1080/03610918.2018.1441171
Asar, Y., and Erisoglu, M. (2016). A new two-parameter estimator to combat multicollinearity. Communications in Statistics - Theory and Methods, 45(2), 555–565. https://doi.org/10.1080/03610926.2013.841089
Atkinson, A. C., and Riani, M. (2000). Robust diagnostic regression analysis. Springer.
Ayinde, K., and Adebayo, A. A. (2020). A new family of robust Liu-type estimators for regression analysis with outliers and multicollinearity. Journal of Applied Statistics, 47(1), 1–17. https://doi.org/10.1080/02664763.2019.1616827
Ayinde, K., Lukman, A. F., and Abdulrahman, A. (2015). Diagnostic measures of influence in classical and robust regression estimators. Journal of Modern Applied Statistical Methods, 14(1), 240–258. https://doi.org/10.22237/jmasm/1430453820
Bagheri, A., and Midi, H. (2009). Diagnostic robust generalized potentials for the detection of influential observations in linear regression. Journal of Applied Statistics, 36(5), 509–520. https://doi.org/10.1080/02664760802320479
Bagheri, A., Midi, H., and Karimi, R. (2010). Collinearity-influential observations in regression: A Monte Carlo study. Statistical Papers, 51(4), 859–871. https://doi.org/10.1007/s00362-008-0158-6
Belsley, D. A., Kuh, E., and Welsch, R. E. (1980). Regression diagnostics: Identifying influential data and sources of collinearity. Wiley.
Chatterjee, S., and Hadi, A. S. (1986). Influential observations, high leverage points, and outliers in linear regression. Statistical Science, 1(3), 379–393. https://doi.org/10.1214/ss/1177013622
Chatterjee, S., and Hadi, A. S. (1988). Sensitivity analysis in linear regression. Wiley.
Cook, R. D. (1977). Detection of influential observation in linear regression. Technometrics, 19(1), 15–18. https://doi.org/10.1080/00401706.1977.10489493
Emami, A., and Emami, M. (2016). Influence diagnostics using Pena’s statistic in ridge regression. Journal of Statistical Computation and Simulation, 86(12), 2343–2357. https://doi.org/10.1080/00949655.2015.1121874
Hoerl, A. E., and Kennard, R. W. (1970). Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12(1), 55–67. https://doi.org/10.1080/00401706.1970.10488634
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, M. (2013). On detecting influential observations in regression models based on robust and biased estimators. Communications in Statistics - Theory and Methods, 42(14), 2544–2557. https://doi.org/10.1080/03610926.2011.651453
Jahufer, M., and Jianbao, W. (2008). Modified ridge regression (MRR) for detecting influential observations in presence of multicollinearity. Pakistan Journal of Statistics, 24(2), 123–132.
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. G. (2003). Performance of some new ridge regression estimators. Communications in Statistics—Simulation and Computation, 32(2), 419–435. https://doi.org/10.1081/SAC-120017499
Lawless, J. F., and Wang, P. (1976). A simulation study of ridge and other regression estimators. Communications in Statistics - Theory and Methods, 5(4), 307–323. https://doi.org/10.1080/03610927608827334
Leiva, V., Sanhueza, A., and Silva, M. (2006). Diagnostics in log-Birnbaum–Saunders regression models with censored data. Computational Statistics & Data Analysis, 51(3), 2214–2231. https://doi.org/10.1016/j.csda.2006.06.003
Liu, K. (1993). A new class of biased estimate in linear regression. Communications in Statistics - Theory and Methods, 22(2), 393–402. https://doi.org/10.1080/03610929308831024
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., and Ayinde, K. (2016). Approximate deletion diagnostic for influential points in two-parameter Liu–Ridge regression. Pakistan Journal of Statistics and Operation Research, 12(4), 581–597. https://doi.org/10.18187/pjsor.v12i4.1493
Lukman, A. F., and Ayinde, K. (2018). Performance of some ridge-type estimators when outliers are present. Sri Lankan Journal of Applied Statistics, 19(4), 299–316. https://doi.org/10.4038/sljastats.v19i4.8005
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.
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.
Midi, H., and Zahari, M. (2007). Detecting multivariate outliers in regression. Statistics in Transition, 8(3), 435–448.
Montgomery, D. C., Peck, E. A., and Vining, G. G. (2012). Introduction to linear regression analysis (5th ed.). Wiley.
Peng, Y., Qian, Z., and Sun, Y. (2016). Generalized standardized corrected Pearson residuals in GLMs. Journal of Statistical Computation and Simulation, 86(7), 1396–1410. https://doi.org/10.1080/00949655.2015.1051435
Rahman, M. A., Haque, M. M., and Akhter, S. (2012). Detection of outlying cases in multiple linear regression using population comparison. Journal of Statistical Research, 46(2), 221–231.
Rao, C. R. (1973). Linear Statistical Inference and its applications, 22: John Wiley and
Sons.
Rousseeuw, P. J., and Leroy, A. M. (1987). Robust regression and outlier detection. Wiley.
Rousseeuw, P. J., and van Zomeren, B. C. (1990). Unmasking multivariate outliers and leverage points. Journal of the American Statistical Association, 85(411), 633–639. https://doi.org/10.2307/2290564
Sakallioglu, S., and Kaciranlar, S. (2008). A new class of biased estimators for the linear regression model. Hacettepe Journal of Mathematics and Statistics, 37(1), 41–49.
Ullah, M. I., Ahmed, N., and Ahmed, S. (2013). Influential observations in Liu regression models. Pakistan Journal of Statistics, 29(3), 343–356.
Uzuke, R. O., and Ezeilo, M. A. (2021). Comparative performance of influence measures under ridge regression. Nigerian Journal of Statistics, 38(1), 85–104.
Walker, A. M., and Birch, H. G. (1988). Ridge regression diagnostics and collinearity. Journal of Econometrics, 37(1), 53–70.
Welsch, R. E. (1982). Influence functions for diagnostics. In Modern Data Analysis (pp. 149–167). Academic Press.
Welsch, R. E., and Kuh, E. (1977). Linear regression diagnostics. NBER Technical Paper Series.
Yasin, M., and Murat, A. (2016). Ridge-type estimators under collinearity and outlier conditions. Hacettepe Journal of Mathematics and Statistics, 45(3), 781–795.
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