Volume 8, Issue 5, September 2019, Page: 157-168
Adaptive Partially Linear Regression Models by Mixing Different Estimates
Magda Mohamed Mohamed Haggag, Department of Statistics, Mathematics, and Insurance, Faculty of Commerce, Damanhour University, Damanhour, Egypt
Received: Jun. 24, 2019;       Accepted: Jul. 26, 2019;       Published: Sep. 4, 2019
DOI: 10.11648/j.ajtas.20190805.11      View  544      Downloads  120
This paper proposes adapting the semiparametric partial model (PLM) by mixing different estimation procedures defined under different conditions. Choosing an estimation method of PLM, from several estimation methods, is an important issue, which depends on the performance of the method and the properties of the resulting estimators. Practically, it is difficult to assign the conditions which give the best estimation procedure for the data at hand, so adaptive procedure is needed. Kernel smoothing, spline smoothing, and difference based methods are different estimation procedures used to estimate the partially linear model. Some of these methods will be used in adapting the PLM by mixing. The adapted proposed estimator is found to be a square root-consistent and has asymptotic normal distribution for the parametric component of the model. Simulation studies with different settings, and real data are used to evaluate the proposed adaptive estimator. Correlated and non-correlated regressors are used for the parametric components of the semiparametric partial model (PLM). Best results are obtained in the case of correlated regressors than in the non-correlated ones. The proposed adaptive estimator is compared to the candidate model estimators used in mixing. Best results are obtained in the form of less risk error and less convergence rate for the proposed adaptive partial linear model (PLM).
Backfitting Method, Combining Regression Procedures, Difference Based Method, Partially Linear Models, Profile Likelihood Method, Semiparametric Regression, Spline Smoothing
To cite this article
Magda Mohamed Mohamed Haggag, Adaptive Partially Linear Regression Models by Mixing Different Estimates, American Journal of Theoretical and Applied Statistics. Vol. 8, No. 5, 2019, pp. 157-168. doi: 10.11648/j.ajtas.20190805.11
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Berndt, E. R. (1991). The practice of econometrics, The: classic and contemporary, Addison-Wesley Pub. Co.
Buja, A., Hastie, T. J., and Tibshirani, R. j. (1989). “Linear smoother and additive models (with discussion), Annals of Statistics, 17, 453-555.
Castillo, E.; Castillo, C.; and Hadi, A. S. (2009). “Combining estimates in regression and classification”, Journal of the American Statistical Association, 91, 1641-1650.
Denby, L. (1986). “Smooth regression function”, Statistical Research Report 26. AT&T, Bell Laboratories, Princeton, New Jersy.
Eilers, P. H. C.; and Marx, B. D. 0 (1996). “Flexible smoothing with b-splines and penalities”, Statistical Science, 11, 89–121.
Engle, R.; Granger, C.; Rice, J.; and Weiss, A. (1986). “Nonparametric estimates of the relation between weather and electricity sales”, Journal of the American Statistical Association, 81, 310-320.
Green, P., and Yandell, B. S. (1985). Semi-parametric Generalized Linear Models. Part of the Lecture Notes in Statistics book series (LNS, volume 32).
Green, P., Jennison, C., and Seheult, A. (1985). “Analysis of field experiments by least squares smoothing”, Journal of the Royal Statistical Society, Series B., 47, 299-315.
Hardle, W., Liang, H., and Gao, J. (2000). Partially linear models, Springer Verlag.
Hardle, W., Muller, M., Sperlich, S., and Werwatz, A. (2004). Nonparametric and Semiparametric Modeling: An Introduction, Springer, New York.
Hastie, T. J., and Tibshirani, R. j. (1990). Generalized additive models, Vol. 43 of Monographs on statistics and applied probability, Chapman and Hall, London.
Heckman, N. E., (1986). “Spline smoothing in a partly linear model”, Journal of the Royal Statistical Society, Series B., 48, 244-248.
Holland, A. (2017). “Penalized spline estimation in the partially linear model”, Journal of Multivariate Analysis, 153, 211-235.
LeBlanc, M., and Tibshirani, R. (1996). “Combining estimates in regression and classification”, Journal of the American Statistical Association, 91, 1641-1650.
Li, Q. (1996).”Semiparametric estimation of partially linear panel data models”, Journal of Econometrics, Volume 71, Issues 1–2, 389-397.
Linton, O. B. (1995). “estimation in semiparametric models: A review”. In: Phillips, P. C. B., Maddala, G. S. (Eds.), a Volume in Honor of C. R. Raw. Blackwell.
Liu, Q. (2010). “Asymptotic Theory for Difference-based Estimator of Partially Linear Models”, Journal of the Japan Statistical Society, 39, 393-406.
Muller, M. (2000). Generalized Partial Linear Models, XploRe –Application Guid, 145-170.
Muller, M. (2001). “Estimation and testing in generalized partial linear models- a comparative study, Statistics and comuting, 11, 299-309.
Nkurnziza, S. (2015). “On combining estimation problems under quadratic loss: A generalization”, Windsor Mathematics Statistics Report, University of Windsor, Ontario, Canada.
O’Sullivan, F. (1986). “A statistical perspective on ill-posed inverse problems (with discussion)”, Statistical Science, 1, 505–527.
Opsomer, and Ruppert, 1999. “A root-n consistent backfitting estimator for semiparametric additive modelling”.
Robinson, P. M. (1988). “Root n-consistent semiparametric regression”, Econometrica, 56, 931-954.
Ruppert; D. Wand M. P.; Carroll, R. J. (2003). Semiparametric regression, New York, Cambridge University press.
Severini, T.; and Wong, W (1992). “Generalized profile likelihood and condtional parametric models”, Annals of Statistics, 20, 1768-1802.
Silverman, B. W. (1985). “Some Aspects of the Spline Smoothing Approach to Non-Parametric Regression Curve Fitting”, Journal of the Royal Statistical Society. Series B Vol. 47, No. 1, 1-52.
Speckman, P. (1988). “Kernel smoothing in partial linear models”, Journal of the Royal Statistical Society, Series B., 50, 413-436.
Wahba, G. (1990). Spline models for observational data, Society for Industrial and applied mathematics (Siam), Philadelphia, Pennsylvania.
Wang L.; Brown, L.; and Cai, T. (2011). “A difference based approach to the semiparametric partial linear model”, Electronic Journal of Statistics, 5, 619-641.
Yang, Y. (1999). “Regression with multiple candidate models: selecting or mixing?” Technical report no. 8, Department of Statistics, Iowa state University.
Yang, Y. (2001). “Adaptive regression by mixing”, Journal of the American Statistical Association, 96, 574-588.
Yatchew, A. (1997). “An Elementary Estimator of the Partial Linear Model, Economics Letters, 57, 135-43.
Yatchew, A. (2003). Semiparametric regression for the applied econometrician, New York, Cambridge University press.
Zhou, S.; Chen, X.; Wolfe, D. A. (1998). “Local asymptotics for regression splines and confidence regions”, Annals of Statistics, 26, 1760-1782.
Browse journals by subject