Theory and Applications of the empirical likelihood and finite mixture model

在林业、金融或其他行业，从业者拥有来自许多相关人群的数据。研究它们的异同是一个具有现实意义的问题。在过去，我们证明了密度比模型（DRM）是一个有效的平台来表征相似性。在DRM下，经验似然（EL）方便地利用来自多个群体的所有数据进行有效的推理。对于由光滑估计函数定义的参数，基于EL的方法具有优雅的大样本性质。对于由非光滑函数定义的参数，这些性质中的一些似乎保持不变，但缺乏理论证明。此时，我们考虑的是人口分位数。它们是具有实际重要性的参数，并且与非光滑估计函数相关联。在DRM下的EL的非光滑参数的理解大样本属性的任务是在这个建议的研究问题之一。一种分布相对于另一种分布的随机优势是金融和经济学中的一个重要概念。必须根据可靠的数据和严格的分析来确立主导地位。目前的方法采用基于经验分布的简单统计，这留下了改进的余地。EL-DRM的组合应该为金融研究人员提供更简单，更有效的推理工具。我们将此任务命名为本提案中的另一个示例研究问题。有限混合在我以前的建议中占据了重要的地位，它仍然在统计研究中占据了很大的领土。EM检验在这里取得了一些进展，但它对于有限回归混合模型和隐马尔可夫模型等其他非常规模型缺乏全面的力量。这一建议相信EM测试可以同样强大。然而，我们必须制定一些细节，使这个想法在应用中发挥作用。这将是一个有大量的工作，EM测试开发的所有这些模型。在混合回归中，人们可能对某些特定解释变量的影响的显著性感兴趣。在这种情况下，研究人员可能会对解释变量感兴趣，即使它们只在少数（尽管不是全部）亚群中具有显著影响。如果子群体的数量是已知的，则可以直接将似然比检验应用于来自回归混合的数据。然而，在实践中，知道回归混合的顺序更多的是一种例外而不是常规。在大多数情况下，由于数据量有限，很难以令人满意的确定性确定混合物的顺序。一个可辩护的假设检验程序必须考虑到这种不确定性。在假设检验中如何处理序不确定性是一个很有意义的问题。该提案计划使用基准类型方法来研究这个问题。

In forestry, finance or other industries, practitioners have data from many related populations. Studying their similarity and difference is a problem of practical importance. In the past, we demonstrated that the density ratio model (DRM) is an effective platform to characterize the similarity. Under DRM, empirical likelihood (EL) conveniently utilizes all data from multiple populations for efficient inference. The EL based methods have elegant large sample properties for the parameters defined by smooth estimating functions. For parameters defined by non-smooth functions, some of these properties seem to stay but are short of theoretical justification. At this moment, we have population quantiles in mind. They are parameters of practical importance and associated with non-smooth estimating functions. The task of understanding large sample properties of the EL under DRM for non-smooth parameters is one of the research problems in this proposal. The stochastic dominance of one distribution over another is an important notion in finance and economics. The dominance has to be established based on reliable data and rigorous analysis. Current approaches employ simple statistics based on empirical distributions which leave rooms for improvement. The EL-DRM combination should provide financial researchers with simpler and more efficient inference tools. We name this task as another example research problem in this proposal. Finite mixtures occupy a significant place in my previous proposals and it still occupies a large territory in statistical research. The EM-test has gained some ground here but it lacks sweeping power for other non-regular models as the finite regression mixture and hidden Markov model. This proposal holds faith that the EM test can be made just as powerful. Yet we must work out some particulars to make the idea work in applications. It will be a lot of work to have EM-test developed for all these models. In regression mixtures, one may be interested in the significance of the effect of some specific explanatory variables. Under this scenario, researchers can be interested in explanatory variables even if they have a significant effect only in a few, though not all, subpopulations. If the number of the subpopulations is known, a likelihood ratio test can be directly applied to data from the regression mixtures. However, knowing the order of the regression mixture is more an exception rather than a routine is in practice. In most cases, with a limited amount of data, it is difficult to determine the order of a mixture with a satisfactory certainty. A defensible hypothesis test procedure must take this uncertainty into consideration. The problem of accommodating the order uncertainty in the hypothesis test is of considerable interest. This proposal plans to study this problem using a fiducial type approach.