Mathematical Sciences: Projective Score Methods and Approximate Conditional Inference

本提案所述研究领域的一个共同主题是，滋扰参数的存在以及消除由估计滋扰参数引起的对感兴趣参数的估计偏差的必要性。必须消除偏差的原因是它可能导致感兴趣参数的最大似然估计器的不一致。这些项目都使用投影的方法来修改感兴趣参数的得分函数，从而减少了由于估计滋扰参数而产生的偏差。减少偏差的好处将转移到更准确的参数估计、具有更好覆盖率的置信度区间和测试统计数据，这些统计数据对于小样本量具有“更接近”其渐近极限分布的分布。我们生活在一个信息丰富的时代，也就是数据。我们对这些数据的统计理解源于建立准确的模型。我们手头的大量数据集表明，简单的模型很少能很好地适应数据，因此我们被引导去考虑具有许多未知数的更复杂的模型。拟议的研究将解决涉及许多变量的这些丰富的统计模型所涉及的实际实施和解释问题。

A common theme to the areas of research described in this proposal is the presence of nuisance parameters and the necessity of removing an estimation bias in the parameter of interest, induced by the estimation of the nuisance parameters. The reason why the bias must be removed is that it may lead to inconsistency in the maximum likelihood estimator of the parameter of interest. The projects all use a projective approach to modify the score function for the parameter of interest, so that the bias attributable to the estimation of the nuisance parameter is reduced. The benefits of reducing bias will transfer to more accurate parameter estimates, confidence intervals with better coverage rates and test statistics that for small sample sizes have distributions "closer" to their asymptotic limiting distributions. We live in an age where information, that is data, is abundant. Our statistical understanding of this data arises from making accurate models. The large data sets that we have at hand show that simple models rarely fit the data well, and so we are led to consider more complex models with many unknowns. The proposed research will address the practical implementation and interpretative issues involved with these rich statistical models involving many variables.