Higher order multivariate inference using approximation methods

The bivariate con dence region of two parameters of interest conditioning on one nuisance parameter is evaluated to the error of Op(1/n) using Laplace's approximation. A joint matching prior in the Bayesian framework is developed using Laplace's approximation to establish a mathematical equivalence such that the posterior quartiles coincide with the con dence region de ned as in a frequency approach. Quadratic saddlepoint approximation is developed to evaluate bivariate conditional and unconditional tail probability to the error of Op( 1/n) using saddlepoint approximation. The quadratic saddlepoint approximation is extended to three dimension to evaluate trivariate unconditional tail probability.Ph.D.Includes bibliographical referencesby Yaoshi W