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In the framework of generalized linear models, the nonrobustness of classical estimators and tests for the parameters is a well known problem and alternative methods have been proposed in the literature. These methods are robust and can cope with deviations from the assumed distribution. However, they are based on ̄rst order asymptotic theory and their accuracy in moderate to small samples is still an open question. In this paper we propose a test statistic which combines robustness and good accuracy for moderate to small sample sizes. We combine results from Cantoni and Ronchetti (2001) and Robinson, Ronchetti and Young (2003) to obtain a robust test statistic for hypothesis testing and variable selection which is asymptotically Â2¡distributed as the three classical tests but with a relative error of order O(n¡1). This leads to reliable inference in the presence of small deviations from the assumed model distribution and to accurate testing and variable selection even in moderate to small samples.
Classical inference in statistic and econometric models is typically carried out by means of asymptotic approximations to the sampling distribution of estimators and test statistics. These approximations often do not provide accurate p-values and confidences intervals, especially when the sample size is small. Moreover, even if the sample size is large, the accuracy can be poor due to model misspecification (nonrobustness). Several alternative techniques have been proposed in the statistic and econometric literature to improve the accuracy of clasical inference. In general, these alternatives address either the accuracy of the first-order approximations or the nonrobustness issue. However, the development of general procedures which are both robust and second order accurate is still an open question. In this thesis, we propose an alternative statistical test wich has both robustness and small sample properties for two large and important classes of models: Generalized Linear Models (GLM) and models on overidentifying moments conditions.
This book describes how generalised linear modelling procedures can be used in many different fields, without becoming entangled in problems of statistical inference. The author shows the unity of many of the commonly used models and provides readers with a taste of many different areas, such as survival models, time series, and spatial analysis, and of their unity. As such, this book will appeal to applied statisticians and to scientists having a basic grounding in modern statistics. With many exercises at the end of each chapter, it will equally constitute an excellent text for teaching applied statistics students and non- statistics majors. The reader is assumed to have knowledge of basic statistical principles, whether from a Bayesian, frequentist, or direct likelihood point of view, being familiar at least with the analysis of the simpler normal linear models, regression and ANOVA.