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M-estimation is a widely used technique for robust statistical inference. In this paper, we study model selection and model averaging for M-estimation to simultaneously improve the coverage probability of confidence intervals of the parameters of interest and reduce the impact of heavy-tailed errors or outliers in the response. Under general conditions, we develop robust versions of the focused information criterion and a frequentist model average estimator for M-estimation, and we examine their theoretical properties. In addition, we carry out extensive simulation studies as well as two real examples to assess the performance of our new procedure, and find that the proposed method produces satisfactory results. [ABSTRACT FROM AUTHOR]

In this article, we study model selection and model averaging in quantile regression. Under general conditions, we develop a focused information criterion and a frequentist model average estimator for the parameters in quantile regression model, and examine their theoretical properties. The new procedures provide a robust alternative to the least squares method or likelihood method, and a major advantage of the proposed procedures is that when the variance of random error is infinite, the proposed procedure works beautifully while the least squares method breaks down. A simulation study and a real data example are presented to show that the proposed method performs well with a finite sample and is easy to use in practice. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the model checking problem for a partial linear model when some covariates are missing at random. A weighted model-adjustment method is applied to estimate the regression coefficients and the nonparametric function for the null hypothetical partial linear model. A testing procedure based on a residual-marked empirical process is developed to check the adequacy of the partial linear model. It is shown that the proposed test is consistent and can detect the local alternatives converging to the null hypothetical model at the rate n −1/2. Since the asymptotic null distribution of the testing statistics is case-dependent, an adjusted wild bootstrap method is used to decide the critical value, which is proved to be consistent. A simulation study and a real data analysis are conducted to show that the proposed procedure works well. [ABSTRACT FROM AUTHOR]