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，对数据组合方法（如荟萃分析）的兴趣有所增长。目前的方法通常依赖于强参数假设，但替代方法正在开发中。

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.