Likelihood Inference in Models with a High-Dimensional Nuisance Parameter

似然方法，如最大似然估计和似然比检验，在统计理论和方法中占有重要地位。有大量的工作表明，在相对较弱的条件下，基于似然的推理方法在大样本中是最佳的。这样的结果是基于渐近理论，其中参数的维度保持固定，因为样本量无限增加。然而，基于这种大样本理论的结论可能不适用于参数维度相对于样本量较大的模型。因此，对于实践中使用的许多模型，基于似然推理的标准方法可能表现不佳。本研究的目的是研究和发展基于似然的模型推理方法，其中妨害参数的维度相对于样本量较大。研究将集中在三个广泛的领域：具有地层干扰参数的模型中基于似然统计分布的高阶渐近逼近的发展，具有地层干扰参数的模型的小离散渐近理论的发展，以及具有未知函数的模型的推理方法的发展。该研究将考虑基于似然的推理方法的理论特性以及基于这些结果的新统计方法的发展。统计方法被广泛应用于各个领域。特别是，基于似然的方法已被用于从计算机软件的可靠性到遗传数据的分析等应用中。目前的统计理论大多局限于相对简单的模型，其中可用的数据相对于模型中未知参数的数量很大。然而，在复杂的模型中，可能需要基于相对较少的数据来估计大量的参数。这项研究将为这种类型的模型发展统计理论和方法，这些结果将导致改进的统计方法，这将在许多应用领域有用。这项研究还将进一步加深我们对统计模型性质的理解，因此，将有助于培训统计和相关领域的研究人员。

Likelihood methods, such as maximum likelihood estimation and likelihood ratio tests, play an important role in statistical theory and methodology. There is a large body of work showing that, under relatively weak conditions, likelihood-based methods of inference are optimal in large samples. Such results are based on asymptotic theory in which the dimension of the parameter remains fixed as the sample size increases indefinitely. However, the conclusions based on such a large-sample theory may not be valid for models in which the dimension of the parameter is large relative to the sample size. Thus, for many models used in practice, standard methods of likelihood-based inference may not perform well. The goal of this research is to study and develop likelihood-based methods of inference in models in which the dimension of the nuisance parameter is large relative to the sample size. The research will focus on three broad areas: the development of higher-order asymptotic approximations to the distribution of the likelihood-based statistics in models with stratum nuisance parameters, the development of a small-dispersion asymptotic theory for models with stratum nuisance parameters, and the development of methods of inference in models with an unknown function. The research will consider the theoretical properties of likelihood-based methods of inference as well as the development of new statistical methodology based on those results.Statistical methods are used in a wide range of fields. In particular, likelihood-based methods have been used in in applications ranging from the reliability of computer software to the analysis of genetic data. Much of current statistical theory is restricted to relatively simple models, in which the available data is large relative to the number of unknown parameters in the model. However, in complex models, it may be necessary to estimate a large number of parameters based on relatively little data. This research will develop statistical theory and methodology for this type of model and these results will lead to improved statistical methods that will be useful in many areas of application. The research will also further our understanding of the properties of statistical models and, hence, will be useful in the training of researchers in statistics and related fields.