Asymptotic Approaches to Bayesian and Likelihood Inference

摘要贝叶斯方法和频率论方法是统计推断的两种主要范式。 最近，基于似然性的方法也被提出来解决许多推理问题。 拟议的研究福尔斯落在接口，并认为在这些范式的渐近推理。 高阶渐近展开式在贝叶斯、频率论和似然推理方法中起着重要的作用。 该提案考虑开发此类扩展并将其用于特定应用，例如贝叶斯实验设计，小面积估计和列联表。 小面积估算在许多联邦和地方政府项目中越来越受欢迎。 分层贝叶斯和经验贝叶斯方法都受到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使用无信息先验。 事实上，近年来贝叶斯技术在统计理论和实践中得到了更广泛的接受，部分原因是各种无信息先验。 PI在其过去的研究中使用高阶渐近性（i）开发二阶精确近似值以测量小面积估计中的不确定性，（ii）校准朴素EB置信区间，（iii）比较基于似然性的推断中的各种调整，以及（iv）获得各种无信息先验的频率论验证。 具体地说，将研究以下问题：（a）通过置信集的期望量、点估计的均方误差和其他准则的调整似然的渐近比较;（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）小区域估计中无信息先验的频率主义验证;（e）小区域估计中的最优贝叶斯设计;（f）小区域估计中的最优贝叶斯设计;（g）（e）双向列联表中分数检验零分布的高阶扩展;（f）使用调查权重的小区域估计中的稳健估计近年来，由于联邦和地方政府机构（例如，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）。 一个小区域通常指的是从其中抽取样本的一个人口亚组。 子组可以是地理区域（例如，县）或通过人口因素的交叉分类获得的组。 在许多联邦和地方政府项目中，区域规划和资金分配需要可靠的小区域统计数据。 目前，人口普查局正在对县一级学龄贫困儿童人数进行小面积估计，并为各种地理和人口类别的人口普查计数制定调整因素。 试验设计在工农业生产中起着重要的作用。 最优设计允许实验者在给定预算下获得最大效用。 在定量研究的各个领域，特别是在社会科学中，大量出现的分类数据由实验者感兴趣的各种类别的计数频率组成。 这里要开发的统计解决方案，预计将导致新的和有用的methodolgies在本建议中考虑的研究问题。

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.