Asymptotic inference based on likelihood function

在实践中，关于感兴趣的标量参数的推断通常通过使用一阶渐近方法来实现。 然而，当样本量很小时，这些方法可能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（在尾概率近似方面），但由于这些方法的有限“可访问性”，它们并不受欢迎。 本文的主要目的是针对一些常用的参数模型，将高阶方法应用到R、Splus、Maple、Matlab等标准统计和代数软件中，而对于半参数和非参数模型，高阶方法由于计算复杂，目前还没有得到广泛的应用。 即使是一阶经验似然比方法也经常遇到数值计算问题。 本提案的第二个目的是开发一种方法，该方法可以普遍解决与一阶经验似然比方法相关的数值计算问题，然后“修正”一阶经验似然比方法，以获得更精确的尾概率近似值。贝叶斯推断依赖于假设的先验密度。 先验密度的不同选择对于感兴趣的参数可能具有不同的推断结果。 本建议的第三个目的是应用参数高阶方法的理论结果，以开发一种系统的方法来获得先验密度，该先验密度将具有贝叶斯p值与频率论p值相同的属性。 此外，高阶方法可以应用于获得贝叶斯的p值，而不需要必要的MCMC或高维integrations.The最后一个目的，这个建议是进行系统的比较bootstrap，MCMC，和高阶方法。 特别是，这些方法之间的相似性和差异的研究在尾部概率近似将进行。

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.