Electronic Journal of Statistics

Efficient parameter estimation in regression with missing responses

Ursula U. Müller and Ingrid Van Keilegom

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We discuss efficient estimation in regression models that are defined by a finite-dimensional parametric constraint. This includes a variety of regression models, in particular the basic nonlinear regression model and quasi-likelihood regression. We are interested in the case where responses are missing at random. This is a popular research topic and various methods have been proposed in the literature. However, many of them are complicated and are not shown to be efficient. The method presented here is, in contrast, very simple – we use an estimating equation that does not impute missing responses – and we also prove that it is efficient if an appropriate weight matrix is selected. Finally, we show that this weight matrix can be replaced by a consistent estimator without losing the efficiency property.

Article information

Electron. J. Statist., Volume 6 (2012), 1200-1219.

First available in Project Euclid: 29 June 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators 62G05: Estimation
Secondary: 62J02: General nonlinear regression

Efficiency influence function missing at random nonlinear regression nuisance function parametric regression quantile regression quasi-likelihood regression


Müller, Ursula U.; Van Keilegom, Ingrid. Efficient parameter estimation in regression with missing responses. Electron. J. Statist. 6 (2012), 1200--1219. doi:10.1214/12-EJS708. https://projecteuclid.org/euclid.ejs/1340974141

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  • [1] Bickel P.J. (1982). On adaptive estimation., Ann. Statist., 10, 647-671.
  • [2] Bickel, P.J., Klaassen, C.A.J., Ritov, Y. and Wellner, J. A. (1998)., Efficient and Adaptive Estimation for Semiparametric Models. Springer.
  • [3] Chamberlain, G. (1987). Asymptotic efficiency in estimation with conditional moment restrictions., J. Econometrics, 34, 305-334.
  • [4] Chamberlain, G. (1992). Efficiency bounds for semiparametric regression., Econometrica, 60, 567-596.
  • [5] Chen, X., Linton, O. and Van Keilegom, I. (2003). Estimation of semiparametric models when the criterion function is not smooth., Econometrica, 71, 1591-1608.
  • [6] Forrester, J., Hooper, W., Peng, H. and Schick, A. (2003). On the construction of efficient estimators in semiparametric models., Statist. Decisions, 21, 109-138.
  • [7] Koul, H.L. and Susarla, V. (1983). Adaptive estimation in linear regression., Statist. Decisions, 1, 379-400.
  • [8] Müller, U.U. (2007). Weighted least squares estimators in possibly misspecified nonlinear regression., Metrika, 66, 39-59.
  • [9] Müller, U.U. (2009). Estimating linear functionals in nonlinear regression with responses missing at random., Ann. Statist., 37, 2245-2277.
  • [10] Müller, U.U., Schick, A. and Wefelmeyer, W. (2006). Imputing responses that are not missing. In:, Probability, Statistics and Modelling in Public Health (M. Nikulin, D. Commenges and C. Huber, eds.), 350-363, Springer.
  • [11] Müller, U.U. and Wefelmeyer, W. (2002). Autoregression, estimating functions, and optimality criteria. In:, Advances in Statistics, Combinatorics and Related Areas (C. Gulati, Y.-X. Lin, J. Rayner and S. Mishra, eds.), 180-195, World Scientific Publishing, Singapore.
  • [12] Newey, W.K. (1990). Semiparametric efficiency bounds., J. Appl. Econometrics, 5, 99-135.
  • [13] Newey, W.K. (1993). Efficient estimation of models with conditional moment restrictions. In:, Handbook of Statistics 11: Econometrics (G. S. Maddala, C. R. Rao and H. D. Vinod, eds.), 419-454. Elsevier, Amsterdam.
  • [14] Pakes, A. and Pollard, D. (1989). Simulation and the asymptotics of optimization estimators., Econometrica, 57, 1027-1057.
  • [15] Robins, J.M., Rotnitzky, A. and Zhao, L.P. (1994). Estimation of regression coefficients when some regressors are not always observed., J. Amer. Statist. Assoc., 89, 846-866.
  • [16] Robins, J.M., Rotnitzky, A. and Zhao, L.P. (1995). Analysis of semiparametric regression models for repeated outcomes in the presence of missing data., J. Amer. Statist. Assoc., 90, 106-121.
  • [17] Schick, A. (1987). A note on the construction of asymptotically linear estimators., J. Statist. Plann. Inference, 16, 89-105.
  • [18] Schick, A. (1993). On efficient estimation in regression models., Ann. Statist., 21, 1486-1521. Correction and addendum: 23 (1995), 1862-1863.
  • [19] Tsiatis, A.A. (2006)., Semiparametric Theory and Missing Data. Springer.
  • [20] Wang, D. and Chen, S.X. (2009). Empirical likelihood for estimating equations with missing values., Ann. Statist., 37, 490-517.
  • [21] Wang, Q. and Sun, Z. (2007). Estimation in partially linear models with missing response at random., J. Multivariate Anal., 98, 1470-1493.
  • [22] Wang, Y., Shen, J., He, S. and Wang, Q. (2010). Estimation of single index model with missing response at random., J. Statist. Plann. Inference, 140, 1671-1690.
  • [23] Zhou, Y., Wan, A.T.K. and Wang, X. (2008). Estimating equations inference with missing data., J. Amer. Statist. Assoc., 103, 1187-1199.