Iterative stochastic restricted $r-d$ class estimator in generalized linear models: application to binomial, Poisson and negative binomial distributions

Abstract

In this paper, we provide an iterative stochastic restricted $r-d$ (SR-rd) class estimator that incorporates prior and sample information to address the multicollinearity problem. The newly proposed estimator is a manifold estimator that contains various estimators under specific conditions. The new estimator is compared to the maximum likelihood, principal components regression, and $r-d$ class estimators. To assess the performance, two numerical examples and two simulation studies are performed where the scalar mean square error and expected mean square error are the performance evaluation criteria. The analysis results show that the value of $d$ affects the performance of the estimators. The farther the $d$ value is from zero, the better the SR-rd estimator is compared to other estimators, and the SR-rd estimator is a good estimator at the optimal $d$ value.

Keywords

• Stochastic restrictions; shrinkage parameter
• mean square error
• binomial distribution
• Poisson distribution
• negative binomial distribution
• r-d class estimator

References

1. [1] A. Abbasi and M.R. Özkale, The r-k class estimator in generalized linear models applicable with simulation and empirical study using a Poisson and Gamma responses, Hacet. J. Math. Stat. 50 (2), 594-611, 2021.
2. [2] M.N. Akram, M. Amin, A.F. Lukman, and S. Afzal, Principal component ridge type estimator for the inverse Gaussian regression model. J. Stat. Comput. Simul. 92 (10), 2060-2089, 2022.
3. [3] K.C. Arum and F.I. Ugwuowo, Combining principal component and robust ridge estimators in linear regression model with multicollinearity and outlier, Concurr. Comput. Pract. Exp. 34 (10), 6803, 2022.
4. [4] K.C. Arum, F.I. Ugwuowo, H.E. Oranye, T.O. Alakija, T.E. Ugah, and O.C. Asogwa, Combating outliers and multicollinearity in linear regression model using robust Kibria Lukman mixed with principal component estimator, simulation and computation. Sci. Afr. 19 (17), 2023.
5. [5] M.R. Baye and D.F. Parker, Combining ridge and principal component regression: A money demand illustration, Commun. Stat. Theory Methods 13 (2), 197-205, 1984.
6. [6] H. Daojiang, and Y. Wu A stochastic restricted principal components regression estimator in the linear model Sci. World J. 84 (1), 2014.
7. [7] R.A. Farghali, A.F. Lukman and A. Ogunleye, Enhancing model predictions through the fusion of Stein estimator and principal component regression. J. Stat. Comput. Simul. 94 (8), 1760-1775, 2024.
8. [8] T. Gargi and S. Chandra Two-parameter stochastic restricted principal component estimator in linear regression model, Pak. J. Stat. 35 (2), 127-154, 2019.

9. [9] Y.E. Gawdat, A stochastic restricted mixed Liu-Type estimator in logistic regression model, Appl. Math. Sci. 7, 311-322, 2020.
10. [10] X. Jianwen, and H.u. Yang, On the restricted r −k class estimator and the restricted r − d class estimator in linear regression J. Stat. Comput. Simul. 81 (6), 679-691, 2011.
11. [11] F. Kurtoğlu and M.R. Özkale, Restricted ridge estimator in generalized linear models: Monte Carlo simulation studies on Poisson and binomial distributed responses, Commun. Stat. Simul. Comput. 48 (4), 1-28, 2017.
12. [12] F. Kurtoğlu and M.R. Özkale, Restricted Liu estimator in generalized linear models: Monte Carlo simulation studies on gamma and Poisson distributed responses, Hacet. J. Math. Stat. 48 (4), 1191-1218, 2019.
13. [13] A.F. Lukman, K. Ayinde, O. Oludoun, and C. Onate, Combining modified ridge type and principal component regression estimators. Sci. Afr. 9, e00536, 2020.
14. [14] K. Månsson, B.M.G. Kibria and G. Sukur, On ridge estimators for the negative binomial regression model. Econ. Model. 29(2), 178-184 (2012), Econ. Model. 29 (4), 1483-1488, 2012.
15. [15] G.C. McDonald and D.I. Galarneau, A monte carlo evaluation of some ridge-type estimators, J.Am.Stat.Assoc. 70 (350), 407-416, 1975.
16. [16] R.H. Myers, Classical and modern regression with applications. Belmont, CA: Duxbury press, 1990.
17. [17] B.D. Marx, A continuum of principal component generalized linear regressions. Comput. Statist. Data Anal. 13 (4), 385-393, 1992.
18. [18] J.A. Nelder and R. W. M.Wedderburn, Generalized Linear Models, J.R. Statist.Soc.A 135 (3), 370-384, 1972.
19. [19] H. Nyquist, Restricted Estimation of Generalized Linear Models, J. R. Stat. Soc.Ser.C. 40 (1), 133-141, 1991.
20. [20] M.R. Özkale, The r-d class estimator in generalized linear models: applications on gamma, Poisson and binomial distributed responses, J. Stat. Comput. Simul. 89 (4), 615-640, 2019.
21. [21] M.R. Özkale and H. Nyquist, The stochastic restricted ridge estimator in generalized linear models, Stat. Pap. 62 (3) 1421-1460 (2021) 2019.
22. [22] M.R. Özkale, Iterative algorithms of biased estimation methods in binary logistic regression, Stat. Pap. 57, 991-1016, 2016.
23. [23] M.R. Özkale and A.Abbasi Iterative restricted OK estimator in generalized linear models and the selection of tuning parameters via MSE and genetic algorithm, Stat. Pap., 1-62, 2022.
24. [24] M.R. Özkale Principal components regression estimator and a test for the restrictions, Statistics 36 (15), 43(6), 541-551, 2009.
25. [25] E.P. Smith and B.D. Marx, Ill-conditioned information matrices, generalized linear models and estimation of the effects of acid rain, Environmetrics 1 (1), 57-71, 1990.
26. [26] C. Shalini, and N. Sarkar, A restricted r-k class estimator in the mixed regression model with autocorrelated disturbances, Stat. Pap. 57 (2), 429-449, 2016.
27. [27] N. Varathan, and P. Wijekoon, Liu-Type logistic estimator under stochastic linear restrictions, Stat. Pap. Ceylon J. Sci.47(1), 21-34, 2018.
28. [28] P. Wel, De Massaguer P.R., A.D., Zuniga, S.H. Saraiva, Modeling the growth limit of Alicyclobacillus acidoterrestris CRA7152 in apple juice: effect of pH, Brix, temperature and nisin concentration, J. Food Process. Preserv. 35 (4) 509-517, 2011.
29. [29] P. Walter W, Maximum likelihood estimation for the negative binomial dispersion parameter, Biometrics 863-867, 1990.
30. [30] J. Wu and Y. Asar, On the stochastic restricted Liu-type maximum likelihood estimator in logistic regression model. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68 (1), 643-653, 2019.
31. [31] Z. Weibing, and Y. Li, A new stochastic restricted Liu estimator for the logistic regression model Open J. Stat. 8 (1), 25-37, 2018.