New Directions in Quantile-based Modeling and Analysis

分位数作为一种数据描述和分析工具已经在统计学中赢得了一百多年的地位。近年来，为了响应广泛应用领域的需求，对纳入协变量影响和处理多变量数据的分位数建模的研究得到了加速。研究者解决了一个重要但经常被忽视的问题，即在伪贝叶斯方法的分位数回归上的后验推理的有效性，这在文献中已经很流行。研究者对工作似然的选择和先验的选择如何在有限样本问题和渐近理论中发挥各自的作用进行了仔细的调查。研究者研究了一类新的收缩先验作为渐近框架，以了解贝叶斯方法在数据稀疏区域和涉及高维协变量的问题中估计和预测分位数的效率增益。本研究将加深我们对伪后验推理有效性的理解，并为单分位数或多分位数水平的分位数回归提供渐近有效和高效的推理方法。该研究还将促进一个新的伪贝叶斯框架的模型选择超越分位数回归。此外，研究者还研究了多变量数据分位数的新概念。提议的活动将激发分位数建模和贝叶斯推理领域的新想法和批判性思维。即将开发的新见解和新工具将有助于对气候研究、公共卫生和其他科学努力中罕见事件的估计、预测和假设检验。多元分位数的概念将导致一种有效的统计降尺度方法，以便在局部尺度上进行更好的气候预测。拟议的活动将直接参与研究生的学术训练的一部分。研究人员将与其他研究人员和科学家合作，确保研究结果在广泛的科学界得到适当的传播。

Quantile as a data descriptive and analytic tool has earned its place in statistics for over a hundred years. In recent years, research on quantile modeling to incorporate the effect of covariates and to handle multivariate data has accelerated in response to the needs arising from a broad area of applications. The investigator addresses an important but often neglected question on the validity of posterior inference on quantile regression for the pseudo-Bayesian methods that have become popular in the literature. The investigator conducts a careful investigation into how the choice of a working likelihood and the choice of a prior play their respective roles, both in finite-sample problems, and in the asymptotic theory. The investigator studies a new class of shrinking priors as an asymptotic framework to understand the efficiency gains of the Bayesian methods for estimation and prediction of quantiles in data sparse areas and in problems involving high dimensional covariates. The proposed research will deepen our understanding of the validity of pseudo-posterior inference and suggest asymptotically valid and efficient inferential methods for quantile regression at single or multiple quantile levels. The research will also facilitate a new pseudo-Bayesian framework for model selection beyond quantile regression. Furthermore, the investigator studies a new notion of quantile for multivariate data.The proposed activities will stimulate novel ideas and critical thinking in the areas of quantile modeling and Bayesian inference. The new insights and the new tools to be developed will be useful for estimation, prediction, and hypothesis testing regarding rare events in climate research, public health, and other scientific endeavors. The notion of multivariate quantiles will lead to an efficient statistical downscaling method for better climate projections at localized scales. The proposed activities will engage graduate students directly as part of their academic training. The investigator will work with other researchers and scientists to ensure that the research results are disseminated appropriately to the broad scientific community.