Abstract
In this paper, we consider the partial linear model with the covariables missing at random. A model calibration approach and a weighting approach are developed to define the estimators of the parametric and nonparametric parts in the partial linear model, respectively. It is shown that the estimators for the parametric part are asymptotically normal and the estimators of g(·) converge to g(·) with an optimal convergent rate. Also, a comparison between the proposed estimators and the complete case estimator is made. A simulation study is conducted to compare the finite sample behaviors of these estimators based on bias and standard error.
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References
Ahn H., Powell J.L. (1993). Estimation of censored selection models with a nonparametric selection mechanism. Journal of Econometrics, 58, 3–30
Chen H. (1988). Convergent rates for parametric components in a partly linear model. The Annals of Statistics, 16, 136–146
Chen H., Shiau H.J. (1991). A two-stage spline smoothing method for partially linear model. Journal of Statistical Planning and Inference, 27, 187–201
Engle R., Granger C., Rice J., Weiss A. (1986). Nonparametric estimates of the relation between weather and electricity sales. Journal of the American Statistical Association, 81, 310–320
Gray R. (1994). Spline-based tests in survival analysis. Biometrics, 50, 640–652
Green P., Siverman B.W. (1994). Nonparametric regression and generalized linear model: a rougher’s penalty approach. vol 58 of Monographs on Statistics and Applied Probabilities. New York, Chapman and Hall
Hamilton S.A., Truong Y.K. (1997). Local linear estimation in partly linear models. Journal of Multivariate Analysis, 60, 1–19
Härdle W., Liang H., Gao J. (2000). Partially linear models. Heidelberg, Physica-Verlag
Heckman N. (1986). Spline smoothing in partly linear models. Journal of the Royal Statistical Society B, 48, 244–248
Hong S.-Y. (1999). Automatic bandwidth choice in a semiparametric regression model. Statistica Sinica, 9, 775–794
Horvitz D.G., Thompson D.J. (1952). A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 90, 141–150
Ibrahim J.G. (1990). Incomplete data in generalized linear models. Journal of the American Statistical Association, 85, 765–769
Liang H., Wang S., Robins J.M., Carroll R.J. (2004). Estimation in partially linear models with missing covariates. Journal of the American Statistical Association. 99, 357–367
Lipsitz S.R., Ibrahim J.G. (1996). A conditional model for incomplete covariates in parametric regression models. Biometrika, 83, 916–922
Lipsitz S.R., Ibrahim J.G., Zhao L.P. (1999). A weighted estimating equation for missing covariate data with properties similar to maximum likelihood. Journal of the American Statistical Asssociation, 94, 1147–1160
Little R.J.A., Rubin D.B. (1987). Statistical analysis with missing data. New York, Wiley
Newey W.K., Powell J.L., Walker J.R. (1990). Semiparametric estimation of selection models: some empirical results. American Economic Review, 80, 324–328
Prakasa Rao, B. L. S. (1981). Nonparametric functional estimation. Orlanda, Fla.: Academic Press.
Qi L., Wang C.Y., Prentice R.L. (2005). Weighted estimators for proportional hazards regression with missing covariates. Journal of the American Statistical Association, 100, 1250–1263
Rice J. (1986). Concergence rates for partially splined models. Statistics & Probability Letters, 4, 203–208
Robins J.M., Rontnitzky A., Zhao L.P. (1994). Estimation of regression coefficients when some regressors are not always observed. Journal of the Royal Statistical Society, 89, 846–866
Robinson P.M. (1988). Root n-consistent semiparametric regression. Econometrica, 56, 931–954
Schick A. (1996). Root-n-consistent in partly linear regression models. Statistics and Probability Letters, 28, 353–358
Schmalensee R., Stoker T.M. (1999). Household gaseline demand in the United States. Econometrica, 67, 645–642
Severini T.A., Staniswalis J.G. (1994). Quasilikelihood estimation in semiparametric models. Journal of the American Statistical Asssociation, 89, 501–511
Speckman P. (1988). Kernel smoothing in partial linear models. Journal of the Royal Statistical Society Ser B, 50, 413–436
Stock J.H. (1989). Nonparametric policy analysis. Journal of the American Statistical Association, 84, 567–576
Stock, J. H. (1991). Nonparametric policy analysis: an application to estimating hazardous waste cleanup benefits. In Barnett, Powell, Tauchen (Eds.) Nonparametric and semiparametric methods in econometrics and statistics. pp. 77–98. Cambridge, Cambridge University Press.
Stone C.J. (1980). Optimal rates of convergence for nonparametric estimators. The Annals of Statistics, 8, 1348–1360
Vach, W. (1994). Logistic Regression with missing values in the covariates. Lecture Notes in Statistics, vol. 86. Heidelberg: Springer.
Wang C.Y., Wang S., Gutierrez R.G. (1998). Local linear regression for generalized linear models with missing data. The Annals of Statistics, 26, 1028–1050
Wang Q.H. (1999). Estimation of partial linear error-in-variables models with validation data. Journal of Multivariate Analysis, 69, 30–64
Wang Q.H., Jing B.Y. (1999). Empirical likelihood for partial linear model with fixed design. Statistics & Probability Letters, 41, 425–433
Wang Q.H., Rao J.N.K (2002). Empirical likelhood-based inference under imputation for missing response data. The Annals of Statistics, 30(3): 345–358
Wang Q.H., Linton O., Härdle W. (2004). Semiparametric regression analysis with missing response at random. Journal of the American Statistical Association, 99, 334–345
Zhao L.P., Lipsitz S.R., Lew D. (1996). Regression analysis with missing covariate data using estimating equations. Biometrics, 52, 1165–1182
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Wang, QH. Statistical estimation in partial linear models with covariate data missing at random. Ann Inst Stat Math 61, 47–84 (2009). https://doi.org/10.1007/s10463-007-0137-1
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DOI: https://doi.org/10.1007/s10463-007-0137-1