Mean Square Error Performance of the Modified Jackknifed Ridge Predictors in the Linear Mixed Models

Year 2020, Volume: 36 Issue: 3, 400 - 407, 31.12.2020

Özge Kuran

Abstract

The goal of this study is to introduce the modified jackknifed ridge prediction method in the linear mixed models. Then, the matrix mean square error (MMSE) comparisons are done. Finally, a real data analysis is given to observe the behavior of the modified jackknifed ridge predictors.

Keywords

Jackknifed ridge predictors, Linear mixed models, Modified jackknified ridge predictors, Multicollinearity

References

• [1] Laird, N. M., Ware, J. H. 1982. Random-Effects Models for Longitudinal Data. Biometrics, 38, 963–974.
• [2] Longford, N. T. 1993. Random Coefficient Models. Oxford University Press, New York.
• [3] Harville, D. A. 1977. Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems. Journal of the American Statistical Association, 72, 320-340.
• [4] Lindstrom, M. J., Bates, D. M. 1988. Newton-Raphson and EM Algorithms for Linear Mixed-Effects Models for Repeated-Measures Data. Journal of the American Statistical Association, 83, 1014-1022.
• [5] Diggle, P. J., Liang, K-Y., Zeger, S. L. 1994. Analysis of Longitudinal Data. Oxford University Press, New York.
• [6] Gumedze, F. N., Dunne, T. T. 2011. Parameter Estimation and Inference in the Linear Mixed Model. Linear Algebra and Its Application, 435, 1920-1944.
• [7] Henderson, C. R. 1950. Estimation of Genetic Parameters (Abstract). Annals of Mathematical Statistics, 21, 309–310.
• [8] Henderson, C. R., Kempthorne, O., Searle S. R., von Krosig, C. N. 1959. Estimation of Environmental and Genetic Trends from Records Subject to Culling. Biometrics, 15, 192-218.
• [9] Hoerl, A. E., Kennard, R. W. 1970. Ridge Regression Biased Estimation for Nonorthogonal Problems. Technometrics, 12, 55–67.
• [10] Singh, B., Chaubey, Y. P., Dwivedi, T. D. 1986. An Almost Unbiased Ridge Estimator. Sankhya, 48, 342–346.
• [11] Ohtani, K. 1986. On Small Sample Properties of the Almost Unbiased Generalized Ridge Estimator. Communications in Statistics -Theory and Methods, 15, 1571–1578.
• [12] Singh, B., Chaubey, Y. P. 1987. On Some Improved Ridge Estimators. Statistical Papers, 28, 53–67.
• [13] Nomura, B. 1988. On the Almost Unbiased Ridge Regression Estimator. Communications in Statistics - Simulation and Computation, 17, 729–743.
• [14] Nyquist, H. 1988. Applications of the Jackknifed Procedure in Ridge Regression. Computational Statistics & Data Analysis, 6, 177-183.
• [15] Gruber, M. H. J. 1998. Improving Efficiency by Shrinkage: the James–Stein and Ridge Regression Estimators. Marcell Dekker, New York.
• [16] Özkale, M. R. 2008. A Jackknifed Ridge Estimator in the Linear Regression Model with Heteroscedastic or Correlated Errors. Statistical Probability Letters, 78, 3159–3169.
• [17] Batah, F. S. M., Ramanathan, T. V., Gore, S. D. 2008. The Efficiency of Modified Jackknife and Ridge Type Regression Estimators: a Comparison. Surveys in Mathematics and its Applications, 3, 111–122.
• [18] Khurana, M., Chaubey, Y. P., Chandra, S. 2014. Jackknifing the Ridge Regression Estimator: a Revisit. Communications in Statistics -Theory and Methods, 43, 5249–5262.
• [19] Hu, H., Xia, Y. 2013. Jackknifed Liu Estimator in Linear Regression Models. Wuhan University Journal of Natural Sciences, 18, 331-336.
• [20] Wu, J. 2016. A Jackknifed Difference-Based Ridge Estimator in the Partial Linear Model with Correlated Errors. Statistics, 50, 1463-1375.
• [21] Turkan, S., Özel, G. 2016. A New Modified Jackknifed Estimator for the Poisson Regression Model. Journal of Applied Statistics, 46, 1892-1905.
• [22] Jiang, J., Lahiri, P., Wan, S. M. 2002. A Unified Jackknifed Theory for Empirical Best Prediction with M-Estimation. Annals of Statistics, 30, 1782-1810.
• [23] Özkale, M. R., Arıcan, E. 2018. A First-Order Approximated Jackknifed Ridge Estimator in Binary Logistic Regression. Computional Statistics, doi:10.1007/s00180-018-0851-6.
• [24] Liu, X. Q., Hu, P. 2013. General Ridge Predictors in a Mixed Linear Model. Journal of Theoretical and Applied Statistics, 47, 363–378.
• [25] Özkale, M. R., Can, F. 2017. An Evaluation of Ridge Estimator in Linear Mixed Models: An Example from Kidney Failure Data. Journal of Applied Statistics, 44, 2251–2269.
• [26] Özkale, M. R., Özge, K. 2019. Adaptation of the Jackknifed Ridge Methods to the Linear Mixed Models. Journal of Statistical Computation and Simulation, 89, 3413–3452.
• [27] Yang, H., Ye, H., Xue, K. 2014. A Further Study of Predictions in Linear Mixed Models. Communications in Statistics -Theory and Methods, 43, 4241–4252.
• [28] Pereira, L. N., Coelho, P. S. 2012. A Small Area Predictor under Area-Level Linear Mixed Models with Restrictions. Communications in Statistics -Theory and Methods, 41, 2524-2544.
• [29] Robinson, G. K. 1991. That BLUP is a Good Thing: the Estimation of Random Effects (with Discussion). Statistical Science, 6, 15-51.
• [30] Štulajter, F. 1997. Predictions in nonlinear regression models. Acta Math Univ Comenian, LXVI, 71–81.
• [31] Lawless, J. F., Wang, P. A. 1976. A Simulation Study of Ridge and Other Regression Estimators. Communications in Statistics -Theory and Methods, 5, 307-323.
• [32] Eurostat website, 2018. Greenhouse Gas Emissions by Source Sector (source: EEA, env_air_gge). Available at http://appsso.eurostat.ec.europa.eu/nui/show.do?dataset=env_air_gge&lang=en.
• [33] Kass, R. E., Raftery, A. E. 1995. Bayes Factors. Journal of the American Statistical Association, 90, 773-795.