Dec. 15^th, 2020

 

Abstract

陈松蹊

北京大学

Title: Linear Regression Trees and Forrest

Abstract: This paper proposes a tree-based estimator called linear regression trees (LRT) for segmented linear regression models which prescribe piecewise linear models over unknown domains determined by binary splits of unknown split variables. The proposed estimation is based on an adaptive split variable selection and level estimation algorithm that maximizes, at each internal node, a cumulative conditional Kendall's tau statistic that measures the rank dependence between the regressors and the estimated residuals. The algorithm provides a data-driven way for estimating the unknown segments. Theoretical analysis shows that the binary split selection algorithm leads to consistent identification and estimation of both the genuine split variables and the split levels. The split selection algorithm is then applied recursively to generate a tree. To obtain a right-sized tree, we provide a tree pruning algorithm for LRT with adaptively defined cost-complexity measure. We also propose a hypothesis testing based stopping rule to control the partitioning process, which is more efficient than pruning. We demonstrate theoretically that the number of the leaf nodes of the pruned tree converges to the underlying number of segments in probability. The practical performance of the proposed approach is evaluated via numerical simulations and case studies on nine well studied datasets. The latter shows advantageous predictive performance of the proposed methods over the CART (Classification And Regression Trees), GUIDE (for Generalized, Unbiased Interaction Detection and Estimation), MARS (Multivariate Adaptive Regression Splines) and Random Forests. This is a joint work with Xiangyu Zheng.

 

朱利平

中国人民大学

题目: 非线性相依关系的度量以及独立性检验

摘要: 本报告将报告三个部分。

一、均值独立：我们引入累积散度的概念，来刻画均值独立。

二、区间分位数独立：我们介绍区间分位数独立的思想，这是对统计独立性这个基本概念的直接推广。

三、分布独立：我们对BKR相关系数进行适当的修改，并介绍投影相关系数来描述两个随机向量之间的非线性相依关系。

 

何辉

北京师范大学

题目: 分枝布朗运动极值的偏差概率

摘要: 分枝布朗运动是统计物理学中的基础模型，其极值理论的研究在偏微分方程、自旋玻璃和广义能量模型、对数相关的高斯场、生物数学等领域的研究中有着重要的应用。在这个报告中，我们会介绍相关的偏差概率的估计的一些进展。

 

许惟钧

北京大学

Title: On convergence of a class of wave equations and their invariant Gibbs measure

Abstract: Motivated by weak universality problems in singular SPDEs, we consider a class of higher order nonlinear approximations to 2D cubic defocusing wave equation starting with their corresponding Gibbs measures. We give a sufficient and almost necessary condition for the convergence of the Gibbs measures within this class of approximation. Interestingly, this condition turns out to be strictly stronger than that for the global convergence of the wave dynamics. The main ingredient in treating these measures is a variational method recently developed by Barashkov and Gubinelli.

 

王若度

加拿大滑铁卢大学

Title: E-values and hypothesis testing

Abstract: Testing statistical hypothesis is usually done in sciences using p-values. Recently, e-values have gained attention as potential alternatives to p-values as measures of uncertainty, significance, and evidence. We discuss three important issues on e-values.

1) Advantages and disadvantages of e-values versus p-values in single, multiple, online and collaborative hypothesis testing.

2) The e-BH procedure, a natural analog of the Benjamini-Hochberg (BH) procedure for false discovery rate (FDR) control. Unlike the usual BH procedure, the e-BH procedure controls the FDR at the desired level for any dependence structure between the e-values, which allows for wide applications.

3) Use e-values to construct a model-free backtest of the Expected Shortfall, the most important risk measure in finance and insurance.

 

傅双双

北京科技大学

Title: Gaussian States as Minimum Uncertainty States

Abstract: Gaussian distributions and Gaussian states are fundamental and ubiquitous in probability theory and quantum theory. In bosonic fields, Gaussian states constitute a rather wide family of states including coherent states, squeezed states, thermal states, etc., and have many classical-like features, which are usually defined from the mathematical perspective in terms of characteristic functions. It is well known that some special Gaussian states, such as coherent states, are minimum uncertainty states for the conventional Heisenberg uncertainty relation involving canonical pair of position and momentum observables. A natural question arises as whether all Gaussian states can be characterized as minimum uncertainty states. In this work, we show that indeed Gaussian states coincide with minimum uncertainty states for an information-theoretic refinement of the conventional uncertainty relation. This characterization puts Gaussian states on a novel basis of physical significance.