Stochastic Restricted Maximum Likelihood Estimator in Logistic Regression Model
Affiliation(s)
1 Postgraduate Institute of Science, University of Peradeniya, Peradeniya, Sri Lanka. 2 Department of Mathematics and Statistics, University of Jaffna, Jaffna, Sri Lanka. 3 Department of Statistics and Computer Science, University of Peradeniya, Peradeniya, Sri Lanka.
1 Postgraduate Institute of Science, University of Peradeniya, Peradeniya, Sri Lanka. 2 Department of Mathematics and Statistics, University of Jaffna, Jaffna, Sri Lanka. 3 Department of Statistics and Computer Science, University of Peradeniya, Peradeniya, Sri Lanka.
ABSTRACT
In the presence of multicollinearity in logistic regression, the variance of the Maximum Likelihood Estimator (MLE) becomes inflated. Siray et al. (2015) [1] proposed a restricted Liu estimator in logistic regression model with exact linear restrictions. However, there are some situations, where the linear restrictions are stochastic. In this paper, we propose a Stochastic Restricted Maximum Likelihood Estimator (SRMLE) for the logistic regression model with stochastic linear restrictions to overcome this issue. Moreover, a Monte Carlo simulation is conducted for comparing the performances of the MLE, Restricted Maximum Likelihood Estimator (RMLE), Ridge Type Logistic Estimator(LRE), Liu Type Logistic Estimator(LLE), and SRMLE for the logistic regression model by using Scalar Mean Squared Error (SMSE).
KEYWORDS
Logistic Regression, Multicollinearity, Stochastic Restricted Maximum Likelihood Estimator, Scalar Mean Squared Error
Logistic Regression, Multicollinearity, Stochastic Restricted Maximum Likelihood Estimator, Scalar Mean Squared Error
Cite this paper
Nagarajah, V. , Wijekoon, P. (2015) Stochastic Restricted Maximum Likelihood Estimator in Logistic Regression Model. Open Journal of Statistics, 5, 837-851. doi: 10.4236/ojs.2015.57082.
Nagarajah, V. , Wijekoon, P. (2015) Stochastic Restricted Maximum Likelihood Estimator in Logistic Regression Model. Open Journal of Statistics, 5, 837-851. doi: 10.4236/ojs.2015.57082.
References
[1] Siray, G.U., Toker, S. and, Kaçiranlar, S. (2015) On the Restricted Liu Estimator in Logistic Regression Model. Communications in Statistics—Simulation and Computation, 44, 217-232. http://dx.doi.org/10.1080/03610918.2013.771742 [2] Hosmer, D.W. and Lemeshow, S. (1989) Applied Logistic Regression. Wiley, New York. [3] Ryan, T.P. (1997) Modern Regression Methods. Wiley, New York. [4] Schaefer, R.L., Roi, L.D. and Wolfe, R.A. (1984) A Ridge Logistic Estimator. Communications in Statistics—Theory and Methods, 13, 99-113. http://dx.doi.org/10.1080/03610928408828664 [5] Aguilera, A.M., Escabias, M. and Valderrama, M.J. (2006) Using Principal Components for Estimating Logistic Regression with High-Dimensional Multicollinear Data. Computational Statistics & Data Analysis, 50, 1905-1924. http://dx.doi.org/10.1016/j.csda.2005.03.011 [6] Nja, M.E., Ogoke, U.P. and Nduka, E.C. (2013) The Logistic Regression Model with a Modified Weight Function. Journal of Statistical and Econometric Method, 2, 161-171. [7] Mansson, G., Kibria, B.M.G. and Shukur, G. (2012) On Liu Estimators for the Logit Regression Model. The Royal Institute of Techonology, Centre of Excellence for Science and Innovation Studies (CESIS), Paper No. 259. http://dx.doi.org/10.1016/j.econmod.2011.11.015 [8] Inan, D. and Erdogan, B.E. (2013) Liu-Type Logistic Estimator. Communications in Statistics—Simulation and Computation, 42, 1578-1586. http://dx.doi.org/10.1080/03610918.2012.667480 [9] Duffy, D.E. and Santner, T.J. (1989) On the Small Sample Prosperities of Norm-Restricted Maximum Likelihood Estimators for Logistic Regression Models. Communications in Statistics—Theory and Methods, 18, 959-980. http://dx.doi.org/10.1080/03610928908829944 [10] Theil, H. and Goldberger, A.S. (1961) On Pure and Mixed Estimation in ECONOMICS. International Economic Review, 2, 65-77. http://dx.doi.org/10.2307/2525589 [11] Liu, K. (1993) A New Class of Biased Estimate in Linear Regression. Communications in Statistics—Theory and Methods, 22, 393-402. http://dx.doi.org/10.1080/03610929308831027 [12] Urgan, N.N. and Tez, M. (2008) Liu Estimator in Logistic Regression When the Data Are Collinear. International Conference on Continuous Optimization and Knowledge-Based Technologies, Linthuania, Selected Papers, Vilnius, 323-327. [13] McDonald, G.C. and Galarneau, D.I. (1975) A Monte Carlo Evaluation of Some Ridge-Type Estimators. Journal of the American Statistical Association, 70, 407-416. http://dx.doi.org/10.1080/01621459.1975.10479882 [14] Rao, C.R. and Toutenburg, H. (1995) Linear Models: Least Squares and Alternatives. 2nd Edition, Springer-Verlag, New York, Inc. [15] Rao, C.R., Toutenburg, H., Shalabh and Heumann, C. (2008) Linear Models and Generalizations. Springer, Berlin.
[1] Siray, G.U., Toker, S. and, Kaçiranlar, S. (2015) On the Restricted Liu Estimator in Logistic Regression Model. Communications in Statistics—Simulation and Computation, 44, 217-232. http://dx.doi.org/10.1080/03610918.2013.771742 [2] Hosmer, D.W. and Lemeshow, S. (1989) Applied Logistic Regression. Wiley, New York. [3] Ryan, T.P. (1997) Modern Regression Methods. Wiley, New York. [4] Schaefer, R.L., Roi, L.D. and Wolfe, R.A. (1984) A Ridge Logistic Estimator. Communications in Statistics—Theory and Methods, 13, 99-113. http://dx.doi.org/10.1080/03610928408828664 [5] Aguilera, A.M., Escabias, M. and Valderrama, M.J. (2006) Using Principal Components for Estimating Logistic Regression with High-Dimensional Multicollinear Data. Computational Statistics & Data Analysis, 50, 1905-1924. http://dx.doi.org/10.1016/j.csda.2005.03.011 [6] Nja, M.E., Ogoke, U.P. and Nduka, E.C. (2013) The Logistic Regression Model with a Modified Weight Function. Journal of Statistical and Econometric Method, 2, 161-171. [7] Mansson, G., Kibria, B.M.G. and Shukur, G. (2012) On Liu Estimators for the Logit Regression Model. The Royal Institute of Techonology, Centre of Excellence for Science and Innovation Studies (CESIS), Paper No. 259. http://dx.doi.org/10.1016/j.econmod.2011.11.015 [8] Inan, D. and Erdogan, B.E. (2013) Liu-Type Logistic Estimator. Communications in Statistics—Simulation and Computation, 42, 1578-1586. http://dx.doi.org/10.1080/03610918.2012.667480 [9] Duffy, D.E. and Santner, T.J. (1989) On the Small Sample Prosperities of Norm-Restricted Maximum Likelihood Estimators for Logistic Regression Models. Communications in Statistics—Theory and Methods, 18, 959-980. http://dx.doi.org/10.1080/03610928908829944 [10] Theil, H. and Goldberger, A.S. (1961) On Pure and Mixed Estimation in ECONOMICS. International Economic Review, 2, 65-77. http://dx.doi.org/10.2307/2525589 [11] Liu, K. (1993) A New Class of Biased Estimate in Linear Regression. Communications in Statistics—Theory and Methods, 22, 393-402. http://dx.doi.org/10.1080/03610929308831027 [12] Urgan, N.N. and Tez, M. (2008) Liu Estimator in Logistic Regression When the Data Are Collinear. International Conference on Continuous Optimization and Knowledge-Based Technologies, Linthuania, Selected Papers, Vilnius, 323-327. [13] McDonald, G.C. and Galarneau, D.I. (1975) A Monte Carlo Evaluation of Some Ridge-Type Estimators. Journal of the American Statistical Association, 70, 407-416. http://dx.doi.org/10.1080/01621459.1975.10479882 [14] Rao, C.R. and Toutenburg, H. (1995) Linear Models: Least Squares and Alternatives. 2nd Edition, Springer-Verlag, New York, Inc. [15] Rao, C.R., Toutenburg, H., Shalabh and Heumann, C. (2008) Linear Models and Generalizations. Springer, Berlin.