Generalized Inference in Linear Regression Model

In this thesis, we consider inference problems in linear regression under both homoscedasticity and heteroscedasticity of the error noise. Namely, we construct generalized confidence regions and generalized confidence intervals for regression coefficients of linear regression models. Regressor variables are considered non-stochastic. Independent normal errors with zero mean and constant or varying dispersion are considered. The regression data from two different regimes are considered. In testing the equality of the regression coefficients in the two regimes under heteroscedasticity, we develop the generalized pivotal quantities of their differences and the generalized p-values. Generalized methods of inference are especially useful in multiparameter cases where nontrivial tests are difficult to obtain. We propose generalized test variables and generalized p-values to test the equality of the sets of regression coefficients of the two regimes. The test can be applied efficiently for all sample sizes and for homoscedastic as well as heteroscedastic cases. The simulation study shows that the proposed method preserves the nominal significance level and maintain satisfactory power under heteroscedasticity, and for small and moderate sample sizes. We also construct the generalized confidence region for the difference of the two sets of regression coefficients. When the regression coefficients remained the same for the two regimes under heteroscedasticity, we propose generalized confidence regions and generalized confidence intervals for the regression parameters.;We applied the proposed method on the community health study data of Sarnia in 2005 and the US gasoline consumption data before and after the 1973 oil crisis. The analysis results show that, for both data sets, the regime change is statistically significant at 5% level.