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。研究它们的相同点和不同点是一个具有重要现实意义的问题。过去，我们证明了密度比模型(DRM)是刻画相似性的有效平台。在DRM下，经验似然(EL)可以方便地利用来自多个总体的所有数据进行有效的推断。基于EL的方法对于光滑估计函数定义的参数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是保留的，但缺乏理论上的合理性。此时此刻，我们心中已经有了人口分位数。它们是具有实际意义的参数，与非光滑估计函数相关。理解非光滑参数DRM下EL的大样本性质是该方案的研究问题之一。这种优势必须建立在可靠的数据和严格的分析基础上。目前的方法使用基于经验分布的简单统计，这留下了改进的空间。EL-DRM组合应该会为金融研究人员提供更简单、更有效的推理工具。我们将这一任务称为本建议中的另一个示例研究问题。*有限混合在我之前的建议中占据了重要的位置，它在统计研究中仍然占据着很大的领土。EM-检验在这方面取得了一些进展，但对于其他非规则模型，如有限回归混合模型和隐马尔可夫模型，它缺乏广泛的能力。这项提议相信，新兴市场测试也可以变得同样强大。然而，我们必须制定出一些细节，才能使这个想法在应用中发挥作用。为所有这些模型开发EM检验将是一项大量的工作。*在回归混合中，人们可能对某些特定解释变量的影响的重要性感兴趣。在这种情况下，研究人员可能会对解释变量感兴趣，即使它们只对少数(尽管不是全部)亚群有显著影响。如果子总体的数量是已知的，则可以直接对来自回归混合的数据进行似然比检验。然而，知道回归混合的顺序更多的是一种例外，而不是一种惯例。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。可辩护的假设检验程序必须将这种不确定性考虑在内。在假设检验中适应阶数不确定性的问题是相当感兴趣的。该提案计划使用基准型方法来研究这一问题。

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.