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Robust location estimators in regression models with covariates and responses missing at random. (English) Zbl 1468.62286

The authors consider the general linear model \[ y=\mu(\mathbf{x})+\epsilon, \] where (generally) it is assumed that the error \(\epsilon\) is independent of \(\mathbf{x}\) and \(\mu\) represents a general linear model.
The paper assumes the situation, quite often met in the practice, when some values among either covariates or responses are missing at random. Instead of applying the simplest naive approach of deleting from the analysis those cases with the missing values, the paper concentrates on the robust estimation of the marginal location parameter. Three approaches are discussed:
Analogy of the Horwitz-Thompson approach [D. G. Horvitz and D. J. Thompson, J. Am. Stat. Assoc. 47, 663–685 (1952; Zbl 0047.38301)], where each observation is weighted according to the inverse of the estimated probability of the dropout.
Augmentation of the inverse probability weighting that prevents against a parametric misspecification of the regression model and/or a dropout probability.
Extending the approach of U. U. Müller [Ann. Stat. 37, No. 5A, 2245–2277 (2009; Zbl 1173.62052)] to a robust setting.

The paper is organized as follows: Section 2 describes some marginal measures of interest to be used. In Section 3, three marginal M-location estimators are presented and their consistency is stated together with that of the preliminary robust scale used in the estimation procedure. A numerical study is carried out in Section 4. An ozone data set is analysed in Section 5, while some concluding remarks are given in Section 6. The supplementary material contains some additional results concerning the asymptotic behavior of the estimators, and the description and results of a complementary numerical experiment based on a partial linear regression model.

MSC:

62G35 Nonparametric robustness
62G08 Nonparametric regression and quantile regression
62J05 Linear regression; mixed models
62G20 Asymptotic properties of nonparametric inference
62D10 Missing data
62P12 Applications of statistics to environmental and related topics
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References:

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