Stochastic Restricted Two-Parameter Estimator in Linear Mixed Measurement Error Models

Authors
Nahid Ganjealivand 1
Fatemeh Ghapani 2
Ali Zaherzadeh 1
Farshin Hormozinejad 1
1 Department of Statistics, Ahvaz Branch, Islamic Azad University, Ahvaz, Iran.
2 Department of Mathematics and Statistics, Shoushtar Branch, Islamic Azad University, Shoushtar, Iran.
10.52547/jirss.20.2.79
Abstract
In this study, the stochastic restricted and unrestricted two-parameter estimators of fixed and random effects are investigated in the linear mixed measurement error models. For this purpose, the asymptotic properties and then the comparisons under the criterion of mean squared error matrix (MSEM) are derived. Furthermore, the proposed methods are used for estimating the biasing parameters. Finally, a real data analysis and a simulation study are provided to evaluate the performance of the proposed estimators.
Keywords
Linear Mixed Measurement Error Model
Two Parameter Estimation
Stochastic Restricted Two Parameter Estimation
Mean-square Error Matrix
  1. Eliot, M. N., J. Ferguson, M. P. Reilly, and A. S. Foulkes. 2011. Ridge regression for longitudinal biomarker data, Int. J. Biostat. 7: 1–11. Farebrother, R. W. 1976. Further results on the mean square error of ridge Regression. J. Roy. Stat. Soc. B 38:248–250. Fung, W. K., X. P. Zhong, and B. C. Wei. 2003. On estimation and influence diagnostics in linear mixed measurement error models. American Journal of Mathematical and Management Sciences 23 (1-2):37–59. Ghapani, F., and B. Babadi. 2020. Two parameter weighted mixed estimator in linear easurement error models. Communications in Statistics - Simulation and Computation. https://doi.org/10.1080/03610918 .2020.1825736 Ghapani, F. 2019. Stochastic restricted Liu estimator in linear mixed measurement error models. Communications in Statistics - Simulation and Computation. https://doi.org/10.1080/03610918 .2019.1664581. Gilmour, A. R., B. R. Cullis, S. J. Welham, B. J. Gogel, and R. Thompson. 2004. An efficient computing strategy for predicting in mixed linear models, Comput. Statist. Data Anal. 44 :571–586. Güler, H., and S. Kaçıranlar. 2009. A comparison of mixed and ridge estimators of linear models.Communications in Statistics - Simulation and Computation 38 (2):368–401. Harrison, D., and D. L. Rubinfeld. 1978. Hedonic housing prices and the demand for clean air. Journal of Environmental Economics and Management 5 (1):81–102. Hoerl, A. E., and R. W. Kennard. 1970. Ridge regression: biased estimation for non-orthogonal problems. Technometrics, 12:55–67. Jiang, J. 2007. Linear and Generalized Linear Mixed Models and Their Applications. Springer, New York. Kibria, B. M. G. 2003. Performance of some new ridge regression estimators. Commun. Statist. Simul. Computat. 32:419–435. Kuran, O., and M. R. Ozkale. 2016. Gilmour’s approach to mixed and stochastic restricted ridge predictions in linear mixed models. Linear Algebra and Its Applications 508:22–47. Liu, K. 1993. A new class of biased estimate in linear regression, Commun. Statist. Theor. Meth. 22(2):393–402. Liu, X. Q., and P. Hu. 2013. General ridge predictors in a mixed linear model, J. Theor. Appl. Stat. 47:363–378. McDonald, G. C., and D. I. Galarneau. 1975. A monte carlo evaluation of some ridge -type estimators, J. Amer. Statist. Assoc. 70:407–416. Nakamura, T. 1990. Corrected score function for errors-in-variables models: Methodology and application to generalized linear models. Biometrika 77 (1):127–37. Rao, C. R., Toutenburg, H. 1995. Linear Models: Least Squares and Alternatives. New York: Springer-Verlag. Rao, C. R., H. Toutenburg, Shalabh, and C. Heumann. 2008. Linear models and generslizations. Berlin: Springer. Ozkale, M. R., and F. Can. 2017. An evaluation of ridge estimator in linear mixed models: an example from kidney failure data, J. Appl. Statist. 44(12):2251–2269. Ozkale, M. R., and O. Kuran. 2018. A further prediction method in linear mixed models: Liu prediction. Communications in Statistics - Simulation and Computation. 49(12): 3171–3195 Ozkale, M. R., and Kaciranlar, S. 2007. The restricted and unrestricted two-parameter estimators. Commun. Statist. Theor. Meth. 36:2707–2725. Searle, S.R., G. Casella, and C.E. McCulloch. 1992. Variance Components. John Wiley and Sons, New York. Theil, H. 1963. On the use of incomplete prior information in regression analysis. J. Amer. Statist. Assoc. 58:401–414. Theil, H., and A. S. Goldberger. 1961. On pure and mixed statistical estimation in economics. Int. Econ. Rev. 2:65–78 Trenkler, G., and H. Toutenburg. 1990. Mean squared error matrix comparisons between biased estimators—An overview of recent results. Statistical Papers 31 (1):165–79. Wald, A. 1943. Tests of Statistical Hypotheses Concerning Several Parameters When the Number of Observations is Large, Trans. Amer. Math. Soc. 54:426-482. Yang, H., and X. Chang. 2010. A New Two-Parameter Estimator in Linear Regression, Commun. Statist. Theor. Meth.39:923–934. Yavarizadeh, B., A. Rasekh, S. E. Ahmed , and B. Babadi. 2019. Ridge estimation in linear mixed measurement error models with stochastic linear mixed restrictions. Communications in Statistics - Simulation and Computation. https://doi.org/10.1080/03610918.2019.1705974 Zare, K., A. Rasekh, and A. A. Rasekhi. 2012. Estimation of variance components in linear mixed measurement error models. Statistical Papers 53 (4):849–63. Zhong, X. P., W. K. Fung, and B. C. Wei. 2002. Estimation in linear models with random effects and errors-in-variables. Annals of the Institute of Statistical Mathematics 54 (3):59
  2. Eliot, M. N., Ferguson, J. , Reilly, M. P., and Foulkes A. S. (2011), Ridge regression for longitudinal biomarker data. Int. J. Biostat, 7 , 1-11. [DOI:10.2202/1557-4679.1353]
  3. Farebrother, R. W. (1976), Further results on the mean square error of ridge Regression. J. Roy. Stat. Soc. B, 38 , 248-250. [DOI:10.1111/j.2517-6161.1976.tb01588.x]
  4. Fung, W. K., Zhong, X. P., and Wei, B. C. (2003), On estimation and influence diagnostics in linear mixed measurement error models. American Journal of Mathematical and Management Sciences, 23 (1-2), 37-59. [DOI:10.1080/01966324.2003.10737603]
  5. Ghapani, F., and Babadi, B. (2020), Two parameter weighted mixed estimator in linear easurement error models. Communications in Statistics-Simulation and Computation , https://doi.org/10.1080/03610918.2020.1825736 [DOI:10.1080/03610918 .2020.1825736]
  6. Ghapani, F. (2019), Stochastic restricted Liu estimator in linear mixed measurement error models. Communications in Statistics - Simulation and Computation , https://doi.org\/10.1080/03610918 .2019.1664581. [DOI:10.1080/03610918.2019.1664581]
  7. Gilmour, A. R., Cullis, B. R., Welham, S. J., Gogel, B. J., and Thompson, R. (2004), An efficient computing strategy for predicting in mixed linear models, Comput. Statist. Data Anal., 44 , 571-586. [DOI:10.1016/S0167-9473(02)00258-X]
  8. Guler, H., and Kac{c ıranlar, S. (2009), A comparison of mixed and ridge estimators of linear models. Communications in Statistics - Simulation and Computation, 38 (2),368-401. [DOI:10.1080/03610910802506630]
  9. Harrison, D., and Rubinfeld, D. L. (1978), Hedonic housing prices and the demand for clean air. Journal of Environmental Economics and Management, 5 (1), 81-102. [DOI:10.1016/0095-0696(78)90006-2]
  10. Hoerl, A. E., and Kennard, R. W. (1970), Ridge regression: biased estimation for non-orthogonal problems. Technometrics, 12 , 55-67. [DOI:10.1080/00401706.1970.10488634]
  11. Jiang, J. (2007),Linear and Generalized Linear Mixed Models and Their Applications . Springer, New York.
  12. Kibria, B. M. G. (2003), Performance of some new ridge regression estimators. Commun. Statist. Simul. Computat., 32 , 419-435. [DOI:10.1081/SAC-120017499]
  13. Kuran, O., and Ozkale, M. R. (2016), Gilmour's approach to mixed and stochastic restricted ridge predictions in linear mixed models. Linear Algebra and Its Applications, 508 , 22-47. [DOI:10.1016/j.laa.2016.06.040]
  14. Liu, K. (1993). A new class of biased estimate in linear regression. Commun. Statist. Theor. Meth., 22 (2), 393-402. [DOI:10.1080/03610929308831027]
  15. Liu, X. Q., and Hu, P. (2013), General ridge predictors in a mixed linear model. J. Theor. Appl. Stat., 47 , 363-378. [DOI:10.1080/02331888.2011.592190]
  16. McDonald, G. C., and Galarneau, D. I. (1975), A monte carlo evaluation of some ridge -type estimators. J. Amer. Statist. Assoc., 70 , 407-416. [DOI:10.1080/01621459.1975.10479882]
  17. Nakamura, T. (1990), Corrected score function for errors-in-variables models: Metho dology and application to generalized linear models. Biometrika, 77 (1), 127-37. [DOI:10.1093/biomet/77.1.127]
  18. Rao, C. R., Toutenburg, H. (1995), Linear Models: Least Squares and Alternatives . New York: Springer-Verlag. [DOI:10.1007/978-1-4899-0024-1]
  19. Rao, C. R., Toutenburg, H., and Heumann, C. (2008), Linear models and generslizations . Berlin: Springer.
  20. Ozkale, M. R., and Can, F. (2017), An evaluation of ridge estimator in linear mixed models: an example from kidney failure data. J. Appl. Statist., 44 (12), 2251-2269. [DOI:10.1080/02664763.2016.1252732]
  21. Ozkale, M. R., and Kuran, O. (2018), A further prediction method in linear mixed models: Liu prediction. Communications in Statistics - Simulation and Computation, 49 (12), 3171-3195 [DOI:10.1080/03610918.2018.1535071]
  22. Ozkale, M. R., and Kacc iranlar, S. (2007), The restricted and unrestricted two-parameter estimators. Commun. Statist. Theor. Meth., 36 , 2707-2725. [DOI:10.1080/03610920701386877]
  23. Searle, S. R., Casella, G., and McCulloch, C. E. (1992), Variance Components . John Wiley and Sons, New York. [DOI:10.1002/9780470316856]
  24. Theil, H. (1963), On the use of incomplete prior information in regression analysis. J. Amer. Statist. Assoc., 58 , 401-414. [DOI:10.1080/01621459.1963.10500854]
  25. Theil, H., and Goldberger, A. S. (1961), On pure and mixed statistical estimation in economics. Int. Econ. Rev., 2 , 65-78 [DOI:10.2307/2525589]
  26. Trenkler, G., and Toutenburg, H. (1990), Mean squared error matrix comparisons between biased estimators-An overview of recent results. Statistical Papers, 31 (1), 165-79. [DOI:10.1007/BF02924687]
  27. Yang, H., and Chang, X. (2010), A New Two-Parameter Estimator in Linear Regression, Commun. Statist. Theor. Meth., 39 , 923-934. [DOI:10.1080/03610920902807911]
  28. Yavarizadeh, B., Rasekh, A., Ahmed, S. E., and Babadi, B. (2019), Ridge estimation in linear mixed measurement error models with stochastic linear mixed restrictions. Communications in Statistics - Simulation and Computation , [DOI:10.1080/03610918.2019.1705974]
  29. Zare, K., Rasekh, A., and Rasekhi, A. A. (2012), Estimation of variance components in linear mixed measurement error models. Statistical Papers, 53 (4), 849-63. [DOI:10.1007/s00362-011-0387-0]
  30. Zhong, X. P., Fung, W. K., and Wei, B. C. (2002), Estimation in linear models with random effects and errors-in-variables. Annals of the Institute of Statistical Mathematics, 54 (3), 595-606 [DOI:10.1023/A:1022467212133]
Journal of the Iranian Statistical Society
Volume 20, Issue 2
December 2021
Pages 79-102

  • Receive Date 23 July 2022