Statistical Inference in Multivariate Settings

Abstract

Precise and reliable inferences are among one of the main tenets of the statistical practice. The ability to make such inferences in modeling can only be made when collected data satisfies the assumptions of the model chosen for inference. The topics covered in this dissertation are varied, but precise and reliable inference for multiple variables under realistic modeling assumptions is a unifying theme. When data come from a discrete exponential family, an inferential framework is developed for when the maximum likelihood estimator does not exist in the usual sense. Envelope methodology is incorporated with aster models so that expected Darwinian fitnesses can be estimated precisely. A residual bootstrap routine for a weighted envelope estimator which accounts for model selection volatility is developed. A residual bootstrap routine is developed in the context of the multivariate linear regression model. These routines show that the variability of the respective estimators is estimated consistently by bootstrapping. Engineering dimension analysis is extended to the multivariate design of experiments context. Outside of the main theme, a central limit theorem under additive deformations is provided in the last chapter.