**June 20, 2019**. Read More

10.5120/ijca2019918339 |

Taiwo Stephen Fayose and Kayode Ayinde. Different Forms Biasing Parameter for Generalized Ridge Regression Estimator. *International Journal of Computer Applications* 181(37):21-29, January 2019. BibTeX

@article{10.5120/ijca2019918339, author = {Taiwo Stephen Fayose and Kayode Ayinde}, title = {Different Forms Biasing Parameter for Generalized Ridge Regression Estimator}, journal = {International Journal of Computer Applications}, issue_date = {January 2019}, volume = {181}, number = {37}, month = {Jan}, year = {2019}, issn = {0975-8887}, pages = {21-29}, numpages = {9}, url = {http://www.ijcaonline.org/archives/volume181/number37/30274-2019918339}, doi = {10.5120/ijca2019918339}, publisher = {Foundation of Computer Science (FCS), NY, USA}, address = {New York, USA} }

### Abstract

The concept of different forms on the basis of original, minimum, maximum and measures of locations of eigen values of XIX of the design matrix in regression analysis was introduced into estimating the biasing or ridge parameter of the generalized ridge estimator of linear regression model with multicollinearity problem. This resulted into some proposed biasing parameters having considered existing seven (7) biasing parameters of the Generalized Ridge Regression (GRR) estimator. Their performances were examined and compared with the Ordinary Least Square (OLS) estimator and the existing (parent / original) biasing parameters of GGR estimator so as to identify the one(s) that would produce efficient estimates of the model parameters. Monte Carlo experiments were conducted 5000 times on two linear regression models with three and six explanatory (p = 3 and p = 6) variables under six (6) levels of multicollinearity (ρ = 0.8, 0.9, 0.95, 0.99, 0.999, 0.9999), three (3) levels of standard error (σ = 1, 5 and 10) and seven (7) levels of sample sizes (n = 10, 20, 30, 50, 100, 150, 250). The estimators were compared using Mean Square Error (MSE) criterion.

Results showed that the proposed different forms biasing parameters frequently perform more efficiently than the parent form; and that the different form of minimum of eigen values of XIX using the generalized ridge parameter of Batach et al. (2008) often produces efficient estimates of linear regression parameter with multicollinearity problem.

### References

- Abdi, H. (2013). Partial Least Squares (PLS) Regression. In Lewis Beck et al (eds) Encyclopedia of Social Sciences Research Methods. 792-795.
- Alkhamisi, M. and Shukur, G. (2008). Developing Ridge Parameters for SUR Model. Communications in Statistics – Theory and Methods, 37(4), 544-564.
- Batach, F. S., Ramnathan, T. and Gore, S. D. (2008). The Efficiency of Modified Jackknife and Ridge Type Regression Estimators: A Comparison. Surveys in Mathematics and its Applications 24 (2), 157–174.
- Bowerman, B. L. and O’Connell, R. T. (1990). Linear Statistical Models: An Applied Approach (2nd ed.). Boston: PWS-Kent Publishing Company.
- Buonaccorsi, J. P. (1996). A Modified Estimating Equation Approach to Correcting for Measurement Error in Regression, Biometrika 83, 433 – 440.
- Dorugade, A. V. and Kashid, D. N. (2010): Alternative Method for Choosing Ridge Parameter for Regression. Applied Mathematical Sciences, 4(9), 447-456.
- Dorugade, A. V. (2016). New Ridge Parameters for Ridge Regression. Journal of the Association of Arab Universities for Basic and Applied Sciences, 1-6.
- Firinguetti, L. A. (1999). Generalized Ridge Regression Estimator and its Finite Sample Properties. Communications in Statistics. Theory and Methods 28(5), 1217-1229.
- Gibbons, D. G. (1981). A Simulation Study of Some Ridge Estimators. Journal of the American Statistical Association, 76(373), 131-139.
- Gujarati, D. N. (1995). Basic Econometrics, McGraw Hill, New York.
- Gujarati, D. N. (2003). Basic Econometrics, McGraw Hill, New York.
- Hoerl, A. E. and Kennard, R. W. (1970a). Ridge Regression: Biased Estimation for Non- Orthogonal Problems. Technometrics, 12, 55-67.
- Hoerl, A. E. and Kennard, R. W. (1970b). Ridge Regression: Applications to Non-orthogonal Problems. Technometrics 12, 69-82.
- Hoerl, A. E., Kennard, R. W. and Baldwin, K. F. (1975). Ridge Regression: Some Simulations. Communications in Statistics, 4(2), 105-123.
- John, T. M. (1998). Mean Squared Error Properties of the Ridge Regression Estimated Linear Probability Models, Ph.D. Dissertation, University of Delaware.
- Johnston, J. (1987). Econometrics Methods. Mc. Graw-Hill, Auckland.
- Khalaf, G. and Shukur, G. (2005). Choosing Ridge Parameters for Regression Problems. Communications in Statistics – Theory and Methods, 34(5), 1177-1182.
- Kibria, B. M. G. (2003). Performance of Some New RidgeRegression Estimators. Communications in Statistics – Simulation and Computation, 32(2), 419-435.
- Kibria, B. M. G. and Banik, S. (2016). Some Ridge Regression Estimators and Their Performances. Journal of Modern Applied Statistical Methods: 15 (1), 206-238.
- 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.
- Lukman, A. F. and Ayinde, K. (2017). Review and Classifications of the Ridge Parameter Estimation Techniques. Haccetteppe Journal of Mathematics and Statistics, 46 (5), 953-967.
- Maddala, G. S. (2005). Introduction to Econometrics. John Wiley and Sons (Asia). Pte. Ltd. Singapore.
- Mansson, K., Shukur, G. and Kibria, B. M. G. (2010). On Some Ridge Regression Estimators: A Monte Carlo simulation study under different error variances. Journal of Statistics, 17(1), 1-22.
- Massy, W. F. (1965). Principal Components Regression in Exploratory Statistical Research.Journal of the American Statistical Association. 60 (309), 234-256.
- McDonald, G. C. and Galarneau, D. I. (1975). A Monte Carlo Evaluation of Ridge-Type Estimators. Journal of the American Statistical Association, 70(350), 407-416.
- Muniz, G. and Kibria, B. M. G. (2009). On Some Ridge Regression Estimators: An Empirical Comparison. Communications in Statistics – Simulation and Computation, 38(3), 621-630.
- Nomura, M. (1988). On The Almost Unbiased Ridge RegressionEstimation. Communication in Statistics – Simulation and Computation, 17(3), 729-743.
- Saleh, A. K. and Kibria, B. M. G. (1993). Performances of Some New Preliminary Test Ridge Regression Estimators and Their Properties. Communications in Statistics – Theory and Methods 22, 2747–2764.
- Troskie, C. G. and Chalton, D. O. (1996). A Bayesian Estimate for the Constants in Ridge Regression. South African Statistical Journal 30, 119–137.
- Vinod, H. D. and Ullah, A. (1981). Recent Advances in Regression Methods, Marcel Dekker Inc. Publication.
- Wichern, D. and Churchill, G. A. (1978). A Comparison of Ridge Estimators. Technometrics, 20, 301-311.

### Keywords

Different Forms Biasing parameter, Ordinary Least Square Estimator, Generalized Ridge Regression Estimator, Mean Square Error