## The Annals of Statistics

### The transfer principle: A tool for complete case analysis

#### Abstract

This paper gives a general method for deriving limiting distributions of complete case statistics for missing data models from corresponding results for the model where all data are observed. This provides a convenient tool for obtaining the asymptotic behavior of complete case versions of established full data methods without lengthy proofs.

The methodology is illustrated by analyzing three inference procedures for partially linear regression models with responses missing at random. We first show that complete case versions of asymptotically efficient estimators of the slope parameter for the full model are efficient, thereby solving the problem of constructing efficient estimators of the slope parameter for this model. Second, we derive an asymptotically distribution free test for fitting a normal distribution to the errors. Finally, we obtain an asymptotically distribution free test for linearity, that is, for testing that the nonparametric component of these models is a constant. This test is new both when data are fully observed and when data are missing at random.

#### Article information

Source
Ann. Statist., Volume 40, Number 6 (2012), 3031-3049.

Dates
First available in Project Euclid: 8 February 2013

https://projecteuclid.org/euclid.aos/1360332192

Digital Object Identifier
doi:10.1214/12-AOS1061

Mathematical Reviews number (MathSciNet)
MR3097968

Zentralblatt MATH identifier
1296.62040

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 62G05: Estimation 62G10: Hypothesis testing

#### Citation

Koul, Hira L.; Müller, Ursula U.; Schick, Anton. The transfer principle: A tool for complete case analysis. Ann. Statist. 40 (2012), no. 6, 3031--3049. doi:10.1214/12-AOS1061. https://projecteuclid.org/euclid.aos/1360332192

#### References

• Bhattacharya, P. K. and Zhao, P.-L. (1997). Semiparametric inference in a partial linear model. Ann. Statist. 25 244–262.
• Cuzick, J. (1992). Efficient estimates in semiparametric additive regression models with unknown error distribution. Ann. Statist. 20 1129–1136.
• Efromovich, S. (2011). Nonparametric regression with responses missing at random. J. Statist. Plann. Inference 141 3744–3752.
• Forrester, J., Hooper, W., Peng, H. and Schick, A. (2003). On the construction of efficient estimators in semiparametric models. Statist. Decisions 21 109–137.
• González-Manteiga, W. and Pérez-González, A. (2006). Goodness-of-fit tests for linear regression models with missing response data. Canad. J. Statist. 34 149–170.
• Hart, J. D. (1997). Nonparametric Smoothing and Lack-of-Fit Tests. Springer, New York.
• Khmaladze, E. V. and Koul, H. L. (2004). Martingale transforms goodness-of-fit tests in regression models. Ann. Statist. 32 995–1034.
• Khmaladze, E. V. and Koul, H. L. (2009). Goodness-of-fit problem for errors in nonparametric regression: Distribution free approach. Ann. Statist. 37 3165–3185.
• Koul, H. L. (2002). Weighted Empirical Processes in Dynamic Nonlinear Models. Lecture Notes in Statistics 166. Springer, New York.
• Koul, H. L. (2006). Model diagnostics via martingale transforms: A brief review. In Frontiers in Statistics (J. Fan and H. L. Koul, eds.) 183–206. Imp. Coll. Press, London.
• Koul, H. L. and Ni, P. (2004). Minimum distance regression model checking. J. Statist. Plann. Inference 119 109–141.
• Li, X. (2012). Lack-of-fit testing of a regression model with response missing at random. J. Statist. Plann. Inference 142 155–170.
• Little, R. J. A. and Rubin, D. B. (2002). Statistical Analysis with Missing Data, 2nd ed. Wiley, Hoboken, NJ.
• Müller, U. U. (2009). Estimating linear functionals in nonlinear regression with responses missing at random. Ann. Statist. 37 2245–2277.
• 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, New York.
• Müller, U. U., Schick, A. and Wefelmeyer, W. (2012). Estimating the error distribution function in semiparametric additive regression models. J. Statist. Plann. Inference 142 552–566.
• Schick, A. (1993). On efficient estimation in regression models. Ann. Statist. 21 1486–1521.
• Schick, A. (1996). Root-$n$-consistent and efficient estimation in semiparametric additive regression models. Statist. Probab. Lett. 30 45–51.
• Stute, W., Xu, W. L. and Zhu, L. X. (2008). Model diagnosis for parametric regression in high-dimensional spaces. Biometrika 95 451–467.
• Sun, Z. and Wang, Q. (2009). Checking the adequacy of a general linear model with responses missing at random. J. Statist. Plann. Inference 139 3588–3604.
• Sun, Z., Wang, Q. and Dai, P. (2009). Model checking for partially linear models with missing responses at random. J. Multivariate Anal. 100 636–651.