Asymptotic Approaches to Bayesian and Likelihood Inference

摘要贝叶斯方法和频域方法是统计推断的两种主要范式。最近，在许多推理问题中也提出了基于似然的方法。建议的研究落在界面上，并考虑了这些范式中的渐近推理。高阶渐近展开式在贝叶斯推论、频率推论和似然推论中起着重要作用。该建议考虑开发这种扩展，并将其用于具体应用，如贝叶斯实验设计、小区域估计和联想表。小面积估算在许多联邦和地方政府项目中变得越来越流行。无论是层次贝叶斯方法还是经验贝叶斯方法都受到了小区域统计用户的青睐。分层贝叶斯过程在很大程度上依赖于非信息性先验的使用。事实上，近年来贝叶斯技术在理论和统计实践中得到更广泛的接受，部分原因是各种非信息性的先验。PI在过去的研究中使用了高阶渐近性来(I)发展二阶精确近似来测量小区域估计中的不确定性，(Ii)校准朴素的EB可信区间，(Iii)在基于似然的推理中比较各种调整与轮廓似然，以及(Iv)获得对各种非信息性先验的频率验证。具体地说，将研究以下问题：(A)通过期望的置信度集体积、点估计的均方误差和其他准则对调整的似然进行渐近比较；(B)方差分量问题中的最优贝叶斯设计；(C)EB区间估计及其在小区域估计中的应用；(D)在小区域估计的背景下非信息先验的频率验证；(E)双向列联表中分数检验的零分布的高阶展开；近年来，由于联邦和地方政府机构(如美国人口普查局、美国劳工统计局、加拿大统计局、澳大利亚统计局、英国中央统计局)对可靠的小区域统计数据的需求不断增长，对小区域估计的研究受到了极大的关注。小区域通常指的是从人群中抽取样本的一个亚群。子组可以是地理区域(例如，县)或通过人口统计因素的交叉分类而获得的组。在许多联邦和地方政府项目的区域规划和资金分配中，需要可靠的小区域统计数据。目前，人口普查局正致力于编制县级贫困学龄儿童人数的小范围估计数，并制定各种地理和人口类别的人口普查统计的调整系数。实验设计在农业和工业生产中发挥着重要作用。最优设计允许实验者在给定预算的情况下获得最大效用。范畴数据广泛存在于定量研究的各个领域，尤其是在社会科学中，它包括实验者感兴趣的各种类别中的计数频率。预计将在这里制定的统计解决方案将导致对本提案中所审议的研究问题提出新的有用的方法。

Abstract Bayes and frequentist approaches are the two main paradigms for statistical inference. Of late, likelihood-based methods are also being proposed in many inferential problems. The proposed research falls in the interface and considers asymptotic inference in these paradigms. Higher order asymptotic expansions play an important role in Bayes, frequentist and likelihood approaches to inference. This proposal considers developing such expansions and using them in specific applications such as Bayesian experimental design, small area estimation and contingency tables. Small area estimation is becoming increasingly popular in many federal and local government programs. Both hierarchical Bayes and empirical Bayes approaches are receiving favorable attention from the users of small area statistics. Hierarchical Bayes procedures rely to a great extent on the use of noninformative priors. Indeed, the wider acceptance of Bayesian techniques in recent years both in the theory and in the practice of statistics is partly due to various noninformative priors. Higher order asymptotics have been used by the PI in his past research to (i) develop second order accurate approximations to measure of uncertainty in small area estimation, (ii) calibrate naive EB confidence intervals, (iii) compare various adjustments to profile likelihoods in likelihood-based inference, and (iv) obtain frequentist validation of various noninformative priors. Specifically, the following problems will be investigated: (a) Asymptotic comparison of adjusted likelihoods via expected volumes of confidence sets, mean squared errors of point estimates and other criteria; (b) Optimal Bayesian designs in variance components problem; (c) EB interval estimation with applications in small area estimation; (d) Frequentist validation of noninformative priors in the context of small area estimation; (e) Higher order expansion of null distribution of score tests in two-way contingency tables; (f) Robust estimation in small area estimation using survey weights.Research on small area estimation has received considerable attention in recent years due to growing demand for reliable small area statistics by federal and local government agencies (e.g., the U.S. Census Bureau, U.S. Bureau of Labor Statistics, Statistics Canada, Australian Bureau of Statistics, Central Statistical Office of U.K.). A small area usually refers to a subgroup of a population from which samples are drawn. The subgroup may be a geographical region (e.g., county) or a group obtained by cross-classification of demographic factors. Reliable small area statistics are needed in regional planning and fund allocation in many federal and local government programs. Currently, the Census Bureau is engaged in developing small area estimates of number of poor children in school-going age at the county level, and developing adjustment factors to the census counts for various geographic and demographic classes. Experimental designs play an important role in agriculture and industrial productions. Optimal designs allow experimenters to derive maximum utility for a given budget. Categorical data, which occur abundantly in every fields of quantitative study, especially in social sciences, consist of frequency of counts in various categories of interest to an experimenter. Statistical solutions to be developed here are expected to lead to new and useful methodolgies on the research problems considered in this proposal.