Weighted likelihood and other weighted methods in statistics

信息技术的进步使数据的搜索和共享变得非常容易。因此，人们对组合数据的方法(如荟萃分析)的兴趣与日俱增。目前的方法通常依赖于强烈的参数假设，但替代方法正在开发中。 这种替代方案，即加权似然率，提供了一个框架，可以在较弱的假设下组合数据。它的关键组成部分是一组权重，用于确定用于推理的样本的相对重要性。我的博士论文提出了一种非参数方法来确定似然的权重，即最小平均均方误差(MAMSE)权重。这项研究项目将：开发MAMSE权重的改进版本，在新的情况下使用它们(例如，用于ROC曲线的荟萃分析)，并开发基于协变量确定权重的策略。本计划还将制定使用MAMSE权重消除Bootstrap样本中的关联的策略。 作为论文的一部分，将加权似然法推广到多变量排序数据。等级用于分析数据的依赖结构，这种依赖可以完全通过数据背后的系词表达。在本程序中，利用多元排序的加权方法将被发展为：解决并列问题，检验m个样本中的Copula的相等性，以及建立非参数相关图。 该计划的长期目标在于开发一个基于加权方法的框架，其中可以组合不同性质的模型，即使它们不共享相同的参数，也无论它们是贝叶斯模型还是频率模型。

Progresses in information technology make the searching and sharing of data very easy. As a consequence, interest in methods for combining data (e.g. for meta-analyses) has grown. Current methods typically rely on strong parametric assumptions, but alternatives are being developed. Such an alternative, the weighted likelihood, offers a framework where it is possible to combine data under weaker assumptions. Its key component is a set of weights that determines the relative importance of the samples used for inference. My doctoral thesis proposes a nonparametric method to determine the weights of the likelihood, the MAMSE (minimum averaged mean squared error) weights. This research project will: develop improved versions of the MAMSE weights, use them in new situations (e.g. for the meta-analysis of ROC curves), and develop strategies to determine weights based on covariates. A strategy to eliminate ties in bootstrap samples using the MAMSE weights will also be developed in this program. As part of my thesis, the weighted likelihood is extended to multivariate rank data. Ranks are used to analyse the dependence structure of the data, a dependence that can be completely expressed through the copula underlying the data. In this program, weighted methods using multivariate ranks will be developed to: solve the problem of tied ranks, test the equality of the copulas in m samples, and build a nonparametric correlogram. The long-term objective of this program consists in developing a framework based on weighted methods where models of different natures could be combined, even if they do not share the same parameters and whether they be Bayesian or Frequentist.