Asymptotic inference based on likelihood function

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In practice, inference concerning a scalar parameter of interest generally is achieved by using first-order asymptotic methods.  These methods, however, can give misleading inference when the sample size is small.  In recent years, higher-order asymptotic methods have been studied extensively.  Regardless of the extreme accuracy (in terms of tail probability approximations) of the higher-order methods, they are not popular because of the limited "accessibility" of these methods.  The first aim of this propoal is to implement the higher-order methos to some standard statistical and algebraic softwares such as R, Splus, Maple, and Matlab for some commonly used parametric models.For semi-parametric and nonparametric models, higher-order methods have not been popular due to the complexity in calculations.  Even first-order empirical likelihood ratio method frequently encountered numerical calculation problems.  The second aim of this proposal is to develop a method that can generally solve the numerical calculation problems associated to the first-order empirical likelihood ratio method, and then "correct" the first-order empirical likelihood ratio method to obtain more accurate tail probability approximations.Bayesian inference depends on the assumed prior density.  A different choice of the prior density may have different inferential results for the parameter of interest.  The third aim of this proposal is to apply the theoretical results in parametric higher-order methods to develop a systematic way of obtaining a prior density which will have the property that the Bayesian's p-values being the same as the frequentist's p-values.  Moreover, the higher-order methods can be applied to obtain the Bayesian's p-values without the necessary MCMC or high dimensional integrations.The last aim of this proposal is to perform systematic comparisons among bootstrap, MCMC, and higher-order methods.  In particular, a study of the similarities and differences among these methods in tail probability approximations will be carried out.