Asymptotic inference based on likelihood function

在实际应用中，关于感兴趣的标量参数的推理一般是用一阶渐近方法来实现的。然而，这些方法在样本量小的情况下会给出误导性的推断。近年来，高阶渐近方法得到了广泛的研究。尽管高阶方法具有极高的精度(就尾概率近似而言)，但由于这些方法的局限性，它们并不受欢迎。本文的首要目的是将高阶方法应用于一些标准的统计和代数软件，如R，sPlus，Maple，对于半参数和非参数模型，由于计算的复杂性，高阶方法一直不受欢迎。即使是一阶经验似然比方法也经常遇到数值计算问题。本建议的第二个目的是开发一种能够普遍解决与一阶经验似然比方法相关的数值计算问题的方法。贝叶斯推理依赖于假设的先验密度。对于感兴趣的参数，不同的先验密度的选择可能会有不同的推断结果。本建议的第三个目的是应用参数高阶方法中的理论结果来发展一种系统地获得先验密度的方法，该先验密度将具有贝叶斯的p值与频率者的p值相同的特性。此外，该建议的第三个目的是应用参数高阶方法中的理论结果来获得具有贝叶斯的p值与频率者的p值相同的属性的先验密度高阶方法可以在没有必要的MCMC或高维积分的情况下获得贝叶斯的p值。本建议的最后目的是对Bootstrap、MCMC和高阶方法进行系统的比较，特别是研究这些方法在尾概率近似方面的异同。

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.