Integrated Likelihood Functions for Non-Bayesian Inference

处理有害参数的问题是统计理论和方法的一个基本方面，特别是在基于可能性的推理中。从统计模型中消除干扰参数的常用方法包括边际和条件推理以及使用轮廓似然函数。另一种方法是使用积分似然法，其中通过对给定权函数进行积分来从似然函数中消除干扰参数。积分似然率的优点是它们总是可用的，而且与轮廓似然率不同，它们是基于平均而不是最大化的，后者已被证明在许多感兴趣的模型中是更可靠的方法。综合似然方法的主要缺点是必须选择实现该方法所需的权重函数。这项研究的目的是研究整合可能性在非贝叶斯、基于可能性的推理中的使用。其中最重要的方面是构建权函数，以便得到的综合似然函数对于非贝叶斯推理是有用的。研究中考虑的其他主题包括高阶渐近理论的发展，计算算法的发展，与现有方法的比较，高维干扰参数模型的应用，以及方法在实际应用中的应用。本研究开发了一种基于积分似然函数的统计理论和方法的新途径。这些方法被用于几乎所有统计模型的分析和应用的所有领域。特别是，综合似然法已被用于从计算机软件的可靠性到遗传数据分析的各种应用。与其他一些最近开发的方法不同，这些方法需要相当多的高级统计理论背景，而综合似然方法相对容易理解和实施。因此，这项拟议的研究结果对广泛领域的研究人员是有用的。这些结果也加深了我们对统计模型性质的理解，从而对统计及相关领域的研究人员的教育起到了重要作用。

The problem of dealing with nuisance parameters is a fundamental aspect of statistical theory and methodology, particularly in likelihood-based inference. Commonly used approaches to eliminating a nuisance parameter from a statistical model include marginal and conditional inference and the use of the profile likelihood function. An alternative approach is to use an integrated likelihood, in which the nuisance parameter is eliminated from the likelihood function by integration with respect to a given weight function. Integrated likelihoods have the advantage that they are always available and, unlike the profile likelihood, they are based on averaging rather than maximization, which has been shown to be a more reliable approach in many models of interest. The primary drawback of the integrated likelihood approach is that weight function needed for its implementation must be chosen. The goal of this research is to study the use of integrated likelihoods in non-Bayesian, likelihood-based, inference. The most important aspect of this is the construction of the weight function so that the resulting integrated likelihood function is useful for non-Bayesian inference. Other topics considered in the research include development of higher-order asymptotic theory, development of computational algorithms, comparisons with existing methods, applications to models with a high-dimensional nuisance parameter, and the application of the methodologyto models used in practice.This research develops a new approach to statistical theory and methodology, based on the use of an integrated likelihood function. These methods are used in the analysis of virtually all statistical models and in all fieldsof application. In particular, integrated likelihood methods have been used in applications ranging from the reliability of computer software to the analysis of genetic data. In contrast to some other recently-developed methods, which require considerable background in advanced statistical theory, the integrated likelihood approach is relatively straightforward to understand and to implement. Thus, the results of this proposed research are useful for researchers in a wide range of fields. The results also further our understanding of the properties of statistical models and, hence, play an important role in the education of researchers in statistics and related fields.