In this article, robust regression parameter inference in the setting of generalized linear models (GLMs) will be proposed. A parametric robust regression methodology that is robust to violations of the distributional assumptions is able to test hypotheses on the regression coefficients in the misspecified GLM setting. More specifically, it will be demonstrated that with large samples the ordinary normal, gamma and inverse Gaussian regression models can be made robust and provide consistent regression parameter estimates in the misspecified GLM setting. These adjusted regression models furnish the correct type I, II error probabilities, and also the correct coverage probability, for continuous data, as long as the true but unknown underlying distributions have finite second moments.

The parametric robust regression techniques are also applied to the analysis of variance (ANOVA) problems including the one-way, two-way ANOVA structures and the one-way analysis of covariance (ANCOVA) setup. In the ANOVA situations, these adjusted regression models continue to remain asymptotically valid representations of the particular parameters of interest, whatever distributions generate the data.

Abstract I
Acknowledgments II
Contents III
1 Introduction 1
2 Robust Tests Based on the Adjusted Likelihood Function 4
2.1 Model Msspecification …………………………………………………... 4
2.2 Robust Likelihood ……………………………………………………….. 11
2.2.1 Models with a One-Dimensional Parameter …………………….. 11
2.2.2 Models with Nuisance Parameters ………………………………. 14
3 Robust Likelihood Inference for the Regression Parameters in the
Setting of Generalized Linear Models 20
3.1 The Large-Sample Behavior of the Maximum Likelihood (ML)
Estimator in Misspecified Generalized Linear Models …………………. 20
3.1.1 Misspecified Generalized Linear Models ……………………….. 21
3.1.2 Asymptotic Properties of the ML Estimate ……………………... 26
3.2 Asymptotic Behavior of the ML Estimate Based on the Superior Working
Models ………………………………………………………………….. 29
3.3 Making Regression Models Robust ……………………………………... 41
4 Robust Regression Models 47
4.1 Robust Normal Regression ………………………………………………. 47
4.2 Robust Gamma Regression ……………………………………………… 50
4.3 Robust Inverse Gaussian Regression …………………………………….. 52
5 Analysis of Variance Models 55
5.1 One-Way ANOVA ………………………………………………………. 55
5.1.1 Normal Working Models ………………………………………... 56
5.1.2 Gamma Working Models ……………………………………….. 61
5.1.3 Inverse Gaussian Working Models ……………………………… 64
5.2 Two-Way ANOVA ………………………………………………………. 67
5.2.1 Normal Working Models ………………………………………... 68
5.2.2 Gamma Working Models ……………………………………….. 72
5.2.3 Inverse Gaussian Working Models ……………………………… 74
5.3 One-Way ANCOVA ……………………………………………………... 76
5.3.1 Normal Working Models ………………………………………... 77
5.3.2 Gamma Working Models ……………………………………….. 78
5.3.3 Inverse Gaussian Working Models ……………………………… 79
6 Simulation Studies 81
6.1 Regression Models ………………………………………………………. 81
6.2 Analysis of Variance Models ……………………………………………. 89
6.2.1 One-Way ANOVA ……………………………………………… 89
6.2.2 Two-Way ANOVA ……………………………………………… 103
6.2.3 One-Way ANCOVA …………………………………………….. 110
7 A Real Example 117
8 Concluding Remarks 119
References 121
Appendix A Adjusting Matrices 124
Appendix B Adjusted Covariance Matrices 132

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