函数型部分线性复合分位数回归模型的估计（英文）

本文研究函数型部分线性复合分位数回归模型的估计问题.我们采用函数型主成分分析方法分析斜率函数,回归样条逼近非参数函数.在相当宽松的条件下给出斜率函数和非参数函数的收敛速度.最后通过理论模拟和实例分析来评价我们提出的方法.     

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多元非参数分位数回归常常是难于估计的, 为了降低维数同时保持非参数估计的灵活性, 人们常常用单指标的方法模拟响应变量的条件分位数. 本文主要研究单指标分位数回归的变量选择. 以最小化平均损失估计为基础, 我们通过最小化具有SCAD惩罚项的平均损失进行变量选择和参数估计. 在正则条件下, 得到了单指标分位数回归SCAD变量选择的Oracle性质, 给出了SCAD变量选择的计算方法, 并通过模拟研究说明了本文所提方法变量选择的样本性质.     

响应变量删失时函数型部分线性分位数回归模型的估计

最近几年,函数型数据分析的理论和应用飞速发展.在许多实际应用里,响应变量往往存在随机右删失的情况.考虑利用函数型部分线性分位数回归模型来刻画函数型和标量预测量与右删失响应变量之间的关系.基于函数型主成分基函数来逼近未知的斜率函数,通过极小化逆概率加权分位数损失函数得到未知系数的估计量.文章的估计方法容易通过加权分位数回归程序实现.在一定的假设条件下,给出了有限维参数估计量的渐近正态性与斜率函数估计量的收敛速度.最后,通过模拟计算与应用实例证明了所提方法的有效性.     

基于Monte Carlo模拟比较K近邻和局部线性分位数回归

局部线性分位数回归是目前比较流行的非参数分位数回归,其潜在假定待估函数线性光滑.K近邻分位数回归也是非参数分位数回归的重要组成部分,其具有不需待估函数光滑和不同分位点的回归曲线不相交等优点.通过Monte Carlo模拟,比较了两者的估计,得到当待估函数的跳跃点或突变点比较多时,K近邻分位数回归的拟合效果优于局部线性回归.其中模拟的函数是Blocks、Bumps和HeaviSine的函数,它们分别代表跳跃性、波动性、斜率突变性的函数.     

无交叉多指标分位数回归模型中的估计与推断

在过去的30年中分位数回归模型的研究已十分深入.然而在实际的应用场景中,由传统估计方法所得到的分位数回归估计量,经常会在不同分位数水平上出现互相交叉的现象,这给分位数回归模型的实际应用造成了解释和预测上的困难.为解决这个问题,本文提出一种带单调约束的半参数多指标分位数回归模型的研究框架.首先将半参数多指标分位数回归模型与充分降维模型相结合,并利用两者间的联系获得指标估计量的相合估计.之后使用张量积样条方法拟合半参数模型在单调约束条件下的非参数结构.通过数值模拟的方式比较所提方法与现有可行方案所得结果在平均预测误差上的差异,实验结果和实际案例的结果都验证了本文所提出模型的可行性.     

空间半参数变系数部分线性回归中的分位数估计

本文研究了空间数据变系数部分线性回归中的分位数估计. 模型中的参数估计量通过未知系数函数的分段多项式逼近得到, 而未知系数函数的估计量通过将参数估计量代入模型中并通过局部线性逼近得到. 文中推导了未知参数向量估计量的渐近分布, 并建立了未知系数函数估计量在内点及边界点的渐近分布. 通过Monte Carlo 模拟研究了估计量的有限样本性质.     

部分函数型线性可加分位数回归模型

文章结合可加分位数回归模型和函数型线性分位数回归模型,提出了部分函数型线性可加分位数回归模型.我们采用函数型主成分基函数逼近斜率函数,B-样条基函数逼近可加函数,提出了模型的估计方法;在一些基本的假设条件下,给出了斜率函数估计和可加函数估计的收敛速度;最后通过模拟计算和应用实例表明了所提方法的有效性.     

纵向数据下半参数回归模型的统计分析

对于纵向数据下半参数回归模型,基于广义估计方程和一般权函数方法构造了模型中参数分量和非参数分量的估计.在适当的条件下证明了参数估计量具有渐近正态性,并得到了非参数回归函数估计量的最优收敛速度.通过模拟研究说明了所提出的估计量在有限样本下的精确性.     

基于分位数回归的大学生成绩变化实证分析

在介绍分位数回归方法的基础上,讨论了分位数回归模型在研究影响大学生成绩变化因素方面的应用,与最小二乘回归模型对比,分位数回归能有效地描述数据尾端的分布情况.     

一般偏差右删失数据下剩余寿命分位数回归

剩余寿命是刻画个体预期寿命的一个重要度量,对剩余寿命的早期研究主要集中在剩余均值上.然而当总体生存函数偏态或厚尾时剩余均值函数可能不存在,因此统计学者建议用剩余寿命分位数来刻画预期寿命.在完全数据和右删失数据下,剩余寿命分位数的建模和理论已经很完善.但是,在实际的调查研究中经常会遇到偏差抽样数据.例如,临床医学中的左截断数据,流行病学中的病例队列抽样数据,医学大型队列研究中的长度偏差抽样数据等等.忽略抽样偏差会导致参数估计有偏和不合理的推断结果.本文考虑一般偏差右删失数据下剩余寿命分位数回归的统计推断问题.首先,我们提出了一个一般偏差右删失数据下的剩余寿命分位数回归模型,并利用一般估计方程方法对模型中的参数进行了估计.针对已有文献常用的删失变量与协变量独立性假设,本文重点考虑了删失变量依赖于协变量场合.其次,由于估计量的渐近方差中涉及非参密度函数,在估计渐近方差时,本文采用Bootstrap方法.最后,数值模拟显示本文提出的方法有限样本性质表现很好.     

The quantile process under random censoring

In this paper we discuss the asymptotic properties of quantile processes under random censoring. In contrast to most work in this area we prove weak convergence of an appropriately standardized quantile process under the assumption that the quantile regression model is only linear in the region, where the process is investigated. Additionally, we also discuss properties of the quantile process in sparse regression models including quantile processes obtained from the Lasso and adaptive Lasso. The results are derived by a combination of modern empirical process theory, classical martingale methods and a recent result of Kato (2009).     

Box-Cox变换用于分位数回归曲线相交问题的研究

将Box-Cox变换与分位数回归模型相结合(两阶段法),是分位数回归研究领域的一大进步。该法虽然两步都与分位数回归的检验函数紧密结合,但是由于没有利用分位数回归的优良性质,而是引入了中间参变量,因此增加了模型的累进误差,降低了模型精度。更重要的是,两阶段法没有对于分位数回归领域中普遍出现的分位数回归曲线的相交问题给出解决方法。针对这些问题,经研究应该首先确定Box-Cox变换的参数,避免模型中不确定因素的引入,然后对数据进行整体变换并结合分位数检验函数,直接利用分位数回归的优良性质,最终确定分位数回归模型的参数。实例证明,该方法提高了模型的精度,可以有效地解决分位数回归曲线的相交问题。     

Using quantile regression for rate-making

Regression models are popular tools for rate-making in the framework of heterogeneous insurance portfolios; however, the traditional regression methods have some disadvantages particularly their sensitivity to the assumptions which significantly restrict the area of their applications. This paper is devoted to an alternative approach–quantile regression. It is free of some disadvantages of the traditional models. The quality of estimators for the approach described is approximately the same as or sometimes better than that for the traditional regression methods. Moreover, the quantile regression is consistent with the idea of using the distribution quantile for rate-making. This paper provides detailed comparisons between the approaches and it gives the practical example of using the new methodology.     

A Frisch-Newton Algorithm for Sparse Quantile Regression

Recent experience has shown that interior-point methods using a log barrier approach are far superior to classical simplex methods for computing solutions to large parametric quantile regression problems. In many large empirical applications, the design matrix has a very sparse structure. A typical example is the classical fixed-effect model for panel data where the parametric dimension of the model can be quite large, but the number of non-zero elements is quite small. Adopting recent developments in sparse linear algebra we introduce a modified version of the Prisch-Newton algorithm for quantile regression described in Portnoy and Koenker~([28]). The new algorithm substantially reduces the storage (memory) requirements and increases computational speed. The modified algorithm also facilitates the development of nonparametric quantile regression methods. The pseudo design matrices employed in nonparametric quantile regression smoothing are inherently sparse in both the fidelity and roughness penalty components. Exploiting the sparse structure of these problems opens up a whole range of new possibilities for multivariate smoothing on large data sets via ANOVA-type decomposition and partial linear models.     

Bayesian lasso binary quantile regression

In this paper, a Bayesian hierarchical model for variable selection and estimation in the context of binary quantile regression is proposed. Existing approaches to variable selection in a binary classification context are sensitive to outliers, heteroskedasticity or other anomalies of the latent response. The method proposed in this study overcomes these problems in an attractive and straightforward way. A Laplace likelihood and Laplace priors for the regression parameters are proposed and estimated with Bayesian Markov Chain Monte Carlo. The resulting model is equivalent to the frequentist lasso procedure. A conceptional result is that by doing so, the binary regression model is moved from a Gaussian to a full Laplacian framework without sacrificing much computational efficiency. In addition, an efficient Gibbs sampler to estimate the model parameters is proposed that is superior to the Metropolis algorithm that is used in previous studies on Bayesian binary quantile regression. Both the simulation studies and the real data analysis indicate that the proposed method performs well in comparison to the other methods. Moreover, as the base model is binary quantile regression, a much more detailed insight in the effects of the covariates is provided by the approach. An implementation of the lasso procedure for binary quantile regression models is available in the R-package bayesQR.     

删失数据中删失指标随机缺失下回归函数的非参数估计

本文在删失数据中删失指标随机缺失的情况下,运用非参数方法给出了回归函数的两种估计量,给出了估计量的一致收敛速度以及渐近分布,并进一步通过数值模拟验证了所提方法在有限样本下的性质.     

Prediction of extremal precipitation by quantile regression forests: from SNU Multiscale Team

This paper considers the problem of spatio-temporal extreme value prediction of precipitation data. This work presents some methods that predict monthly extremes over the next 20 years corresponding to 0.998 quantile at several stations over a certain region. The proposed methods are based on a novel combination of quantile regression forests and circular transformation. As the core of the methodology, quantile regression forests by combining many decorrelated bootstrapping trees may improve prediction performance, and circular transformation is used for building circular transformed predictors of months, which are put into the quantile regression forests model for prediction. The empirical performance of the proposed methods are evaluated through real data analysis, which demonstrates promising results of the proposed approaches.     

Multi-stage nested classification credibility quantile regression model

In insurance (or in finance) practice, in a regression setting, there are cases where the error distribution is not normal and other cases where the set of data is contaminated due to outlier events. In such cases the classical credibility regression models lead to an unsatisfactory behavior of credibility estimators, and it is more appropriate to use quantile regression instead of the ordinary least squares estimation. However, these quantile credibility models cannot perform effectively when the set of data has nested (hierarchical) structure. This paper develops credibility models for regression quantiles with nested classification as an alternative to Norberg’s (1986) approach of random coefficient regression model with multi-stage nested classification. This paper illustrates two types of applications, one with insurance data and one with Fama/French financial data.     

Non-crossing quantile regression via doubly penalized kernel machine

Quantile regression provides a more complete statistical analysis of the stochastic relationships among random variables. Sometimes quantile regression functions estimated at different orders can cross each other. We propose a new non-crossing quantile regression method using doubly penalized kernel machine (DPKM) which uses heteroscedastic location-scale model as basic model and estimates both location and scale functions simultaneously by kernel machines. The DPKM provides the satisfying solution to estimating non-crossing quantile regression functions when multiple quantiles for high-dimensional data are needed. We also present the model selection method that employs cross validation techniques for choosing the parameters which affect the performance of the DPKM. One real example and two synthetic examples are provided to show the usefulness of the DPKM.