Study On Influence Analysis In Linear Models

Linear model is one of the most widely used models in modern statistics, which has many applications in biology, medicine, economics, management, meteorology and so on. In research on linear models, we have to investigate the rationality of the assumption and the effects of the selected data set on the statistical inference, and this is"statistical diagnostics".Influence analysis is an important research topic in statistical diagnostic, and mainly studies and detects"influential cases"which have great influence on the statistical inference. However, most existing methods behave not very well when more than one influential cases appear in samples. These cases may have"masking"effects on each other and cause a slow algorithm speed or an inaccurate detection result. In this dissertation, we try to study the influential measures and masking effects in the field of influence analysis in linear models, and discuss the corresponding subjects on ridge estimation which is a widely applied biased estimation in linear model.Matrix is a necessary mathematical tool in research on linear model. In Chapter 3, some advance on matrix theory is presented. Conditions and useful properties for judging the universal positive definite matrices are given. One special subset of universal positive definite matrices is introduced, and some further properties of this kind of matrices are discussed. Especially, the well-known Minkowski inequality of two matrices is gerneralized. Further, a new partial ordering is defined and some properties and necessary and sufficient conditions for their existence are discussed.In Chapter 4, a new leverage measure matrix is presented for ordinary linear models. It is shown that the measure is more meaningful in geometry and statistics than the traditional one. Further, the leverage measure under ridge estimation is derived and investigated, and is pointed out to be more simple in form than the method in the literature.In Chapter 5, the canonical form of the traditional leverage measure in ridge regression is proposed. The influence of the ridge parameter on the subset of high leverage points is discussed, with theoretical proof and practical data checking.In Chapter 6, we propose masking measures for subset leverage and subset residual in linear regression, and derive a result showing that joint masking and conditional masking effects are equivalent in essence. Moreover, existence of masking is also studied. We point out that the existence of masking for subset leverage is a necessity in practice, but for subset residual it needs additional conditions.In Chapter 7, we present the multiple potential under ridge estimation which can be used to detect the subset of high leverage points. Based on the subset leverage discussed in Chapter 6, we study the responding results under ridge estimation.