Posterior Inference in Bayesian Quantile Regression with Asymmetric Laplace Likelihood

The paper discusses the asymptotic validity of posterior inference of pseudo‐Bayesian quantile regression methods with complete or censored data when an asymmetric Laplace likelihood is used. The asymmetric Laplace likelihood has a special place in the Bayesian quantile regression framework because the usual quantile regression estimator can be derived as the maximum likelihood estimator under such a model, and this working likelihood enables highly efficient Markov chain Monte Carlo algorithms for posterior sampling. However, it seems to be under‐recognised that the stationary distribution for the resulting posterior does not provide valid posterior inference directly. We demonstrate that a simple adjustment to the covariance matrix of the posterior chain leads to asymptotically valid posterior inference. Our simulation results confirm that the posterior inference, when appropriately adjusted, is an attractive alternative to other asymptotic approximations in quantile regression, especially in the presence of censored data.     

Penalized Regression with Ordinal Predictors

Ordered categorial predictors are a common case in regression modelling. In contrast to the case of ordinal response variables, ordinal predictors have been largely neglected in the literature. In this paper, existing methods are reviewed and the use of penalized regression techniques is proposed. Based on dummy coding two types of penalization are explicitly developed; the first imposes a difference penalty, the second is a ridge type refitting procedure. Also a Bayesian motivation is provided. The concept is generalized to the case of non-normal outcomes within the framework of generalized linear models by applying penalized likelihood estimation. Simulation studies and real world data serve for illustration and to compare the approaches to methods often seen in practice, namely simple linear regression on the group labels and pure dummy coding. Especially the proposed difference penalty turns out to be highly competitive.     

ADMM for Penalized Quantile Regression in Big Data

Traditional linear programming algorithms for quantile regression, for example, the simplex method and the interior point method, work well for data of small to moderate sizes. However, these methods are difficult to generalize to high‐dimensional big data for which penalization is usually necessary. Further, the massive size of contemporary big data calls for the development of large‐scale algorithms on distributed computing platforms. The traditional linear programming algorithms are intrinsically sequential and not suitable for such frameworks. In this paper, we discuss how to use the popular ADMM algorithm to solve large‐scale penalized quantile regression problems. The ADMM algorithm can be easily parallelized and implemented in modern distributed frameworks. Simulation results demonstrate that the ADMM is as accurate as traditional LP algorithms while faster even in the nonparallel case.     

SCAD‐penalized quantile regression for high‐dimensional data analysis and variable selection

The present penalized quantile variable selection methods are only applicable to finite number of predictors or do not have oracle property associated with estimator. This technique is considered as an alternative to ordinary least squares regression in case of the outliers and the heavy‐tailed errors existing in linear models. The variable selection through quantile regression with diverging number of parameters is investigated in this paper. The convergence rate of estimator with smoothly clipped absolute deviation penalty function is also studied. Moreover, the oracle property with proper selection of tuning parameter for quantile regression under certain regularity conditions is also established. In addition, the rank correlation screening method is used to accommodate ultra‐high dimensional data settings. Monte Carlo simulations demonstrate finite performance of the proposed estimator. The results of real data reveal that this approach provides substantially more information as compared with ordinary least squares, conventional quantile regression, and quantile lasso.     

Cross Validation Calibration Based Shrinkage – Revisited for the Construction of Prognostic Indices

The use of shrinkage methods for the construction of prognostic indices has been paid increasing attention in the literature on medical statistics in the last years. One approach for the construction of a shrinkage factor is cross validation calibration as suggested by van H ouwelingen and le C essie (1990). We investigate this approach in more detail. First we try to clarify why shrinkage factors constructed by cross validation calibration tend to be smaller than 1. Second we explain why use of this shrinkage factor can result in an improvement of the average prediction error. Third we investigate the possible gain for constellations relevant in medical research by means of a simulation study, focusing on the dilemma, that the improvement on average has to be paid by distinct deteriorations for some patients. Finally we conclude that it is necessary to rethink the choice of loss functions in constructing prognostic indices before recommendations about the use of shrinkage methods can be made.     

Finite sample inference for quantile regression models

Under minimal assumptions, finite sample confidence bands for quantile regression models can be constructed. These confidence bands are based on the “conditional pivotal property” of estimating equations that quantile regression methods solve and provide valid finite sample inference for linear and nonlinear quantile models with endogenous or exogenous covariates. The confidence regions can be computed using Markov Chain Monte Carlo (MCMC) methods. We illustrate the finite sample procedure through two empirical examples: estimating a heterogeneous demand elasticity and estimating heterogeneous returns to schooling. We find pronounced differences between asymptotic and finite sample confidence regions in cases where the usual asymptotics are suspect.     

Regression toward the mean in effectiveness studies: theoretically possible,not mathematically inevitable

Regression Toward the Mean (RTM) in extreme-group quasi-experiments is a thoroughly theoretical matter, not a simple mathematical necessity, so results of extreme-group before-and-after treatment effectiveness studies should not be doubted simply because of the theoretical possibility of RTM. The estimation of RTM, wherever one has a treatment group with a pretreatment mean off to one side of its population’s mean, requires knowledge of the population mean, standard deviation, and regression slope. RTM ought not be presumed to be operative in studies unless its several theoretical assumptions have been justified and its implications for sample data have been corroborated on that data. RTM can properly be adjusted for only when it has been adequately estimated from the relevant population data.     

Karl Pearson and the Establishment of Mathematical Statistics

At the end of the nineteenth century, the content and practice of statistics underwent a series of transitions that led to its emergence as a highly specialised mathematical discipline. These intellectual and later institutional changes were, in part, brought about by a mathematical-statistical translation of Charles Darwin's redefinition of the biological species as something that could be viewed in terms of populations. Karl Pearson and W.F.R. Weldon's mathematical reconceptualisation of Darwinian biological variation and "statistical" population of species in the 1890s provided the framework within which a major paradigmatic shift occurred in statistical techniques and theory. Weldon's work on the shore crab in Naples and Plymouth from 1892 to 1895 not only brought them into the forefront of ideas of speciation and provided the impetus to Pearson's earliest statistical innovations, but it also led to Pearson shifting his professional interests from having had an established career as a mathematical physicist to developing one as a biometrician. The innovative statistical work Pearson undertook with Weldon in 1892 and later with Francis Galton in 1894 enabled him to lay the foundations of modern mathematical statistics. While Pearson's diverse publications, his establishment of four laboratories and the creation of new academic departments underscore the plurality of his work, the main focus of his life-long career was in the establishment and promulgation of his statistical methodology.     

Application of Shrinkage Techniques in Logistic Regression Analysis: A Case Study

Logistic regression analysis may well be used to develop a predictive model for a dichotomous medical outcome, such as short-term mortality. When the data set is small compared to the number of covariables studied, shrinkage techniques may improve predictions. We compared the performance of three variants of shrinkage techniques: 1) a linear shrinkage factor, which shrinks all coefficients with the same factor; 2) penalized maximum likelihood (or ridge regression), where a penalty factor is added to the likelihood function such that coefficients are shrunk individually according to the variance of each covariable; 3) the Lasso, which shrinks some coefficients to zero by setting a constraint on the sum of the absolute values of the coefficients of standardized covariables.
Logistic regression models were constructed to predict 30-day mortality after acute myocardial infarction. Small data sets were created from a large randomized controlled trial, half of which provided independent validation data. We found that all three shrinkage techniques improved the calibration of predictions compared to the standard maximum likelihood estimates. This study illustrates that shrinkage is a valuable tool to overcome some of the problems of overfitting in medical data.     

Methods for Scalar‐on‐Function Regression

Recent years have seen an explosion of activity in the field of functional data analysis (FDA), in which curves, spectra, images and so on are considered as basic functional data units. A central problem in FDA is how to fit regression models with scalar responses and functional data points as predictors. We review some of the main approaches to this problem, categorising the basic model types as linear, non‐linear and non‐parametric. We discuss publicly available software packages and illustrate some of the procedures by application to a functional magnetic resonance imaging data set.     

Lp空间上线性回归方程回归系数的估计

本文讨论了多元线性回归方程参数的L^p估计问题。首先采取消除方程中参数“。的技术,化参数估计为对回归系数的估计,然后进一步将问题转化为一个单目标数学规划模型,利用能自动搜索最优解的电脑软件,十分方便地求出规划模型的最优解。特别对一元回归方程,在消除参数a0后,还可以直接利用普通一元函数求极值的方法,求出回归系数的估计。     

半参数变系数空间误差回归模型的估计

研究目标：克服半参数变系数回归模型中误差项可能存在的空间相关性问题。研究方法：提出一类新的半参数变系数空间误差回归模型,并构造其截面似然估计。研究发现：在小样本条件下,模型估计量具有良好的表现,其精度随着样本容量的增加而提高;应用该方法分析我国资源禀赋与地方公共品供给之间的相互关系,进一步证实了模型较强的适用性。研究创新：证明了估计量的一致性与渐近正态性,并通过蒙特卡洛模拟考察了估计方法的小样本表现。研究价值：新方法对于其他结构的半/非参数空间计量模型理论研究具有推广价值,其估计技术在经济、管理等学科中具有应用价值。     

回归分析在企业效益中的应用

回归分析是数理统计中的一个重要内容,是利用统计学原理寻求隐藏在随机现象中的统计规律的计算方法和理论,它在各个学科领域以及社会经济各部门都得到广泛应用。运用回归分析建立回归模型,并通过逐步回归求得"最优"结果,利用最优回归模型对规模以上企业效益未来发展进行预测,从而为有关部门的决策提供一定的科学依据。     

Regression to the mode

Abstract  High measurement values often show on average a spontaneous decrease when remeasured under stationary study conditions. This effect is known as "regression to the mean", a phenomenon widely met in biomedical research. In this paper a general formula is derived, which shows that this effect should be better called "regression to the mode". Further it is shown that this effect may depend on the time-spacing of repeated measurements in a stationary population.     

“双低一代”的社会经济特征研究

由于即将退休与刚退休的这一代人拥有低赡养比与低抚养比,本文称其为“双低一代”。本文基于宏观统计数据与微观调查数据,利用分位回归等方法对“双低一代”的社会学特征与经济学特征进行研究,发现其具有人口总量较多、教育背景优良、社会阅历丰富、消费倾向明显等特征。此外,通过对“双低一代”的人力资本存量进行测算,提出充分发挥“双低一代”余热、缓解老龄化问题的政策建议。     

Estimators of Regression Parameters for Truncated and Censored Data

Estimators of parameters in semi-parametric left truncated and right censored regression models are proposed. In contrast to the majority of existing estimators, the proposed estimators do not require the error term of the regression model to have a symmetric distribution. In addition the estimators use asymmetric “trimming” of observations. Consistency and asymptotic normality of the estimators are shown. Finite sample properties are considered in a small simulation study. For the left truncated case, an empirical application illustrates the usefulness of the estimator.     

Two-stage rank estimation of quantile index models

This paper estimates a class of models which satisfy a monotonicity condition on the conditional quantile function of the response variable. This class includes as a special case the monotonic transformation model with the error term satisfying a conditional quantile restriction, thus allowing for very general forms of conditional heteroscedasticity. A two-stage approach is adopted to estimate the relevant parameters. In the first stage the conditional quantile function is estimated nonparametrically by the local polynomial estimator discussed in Chaudhuri (Journal of Multivariate Analysis 39 (1991a) 246–269; Annals of Statistics 19 (1991b) 760–777) and Cavanagh (1996, Preprint). In the second stage, the monotonicity of the quantile function is exploited to estimate the parameters of interest by maximizing a rank-based objective function. The proposed estimator is shown to have desirable asymptotic properties and can then also be used for dimensionality reduction or to estimate the unknown structural function in the context of a transformation model.     

基于分位数回归的中国居民消费研究

本文从经济增长理论及一般均衡分析入手,将居民收入和政府支出引入效用函数,探讨消费、生产及政府行为三者之间的关系,得到消费的动态方程。同时基于该方程利用分位数回归进行实证分析,结果表明,不同消费量下各变量对消费有不同的影响,同时对城镇和农村的影响程度也各不相同。     

Reassessing the dependence between economic growth and financial conditions since 1973

Adrian, Boyarchenko and Giannone ((2019), ABG) adapt quantile regression (QR) methods to examine the relationship between US economic growth and financial conditions. We confirm their empirical findings, using their methodology and their pre-2016 sample. Mindful of the importance of the Covid-19 pandemic, we extend the sample to 2021Q3 and find attenuation of the key estimated coefficients using ABG's empirical methods. Given the pandemic observations, we provide robust QR analysis of dependence based on ranked data and explain the relationship with extant copula modelling methods.     

零膨胀损失次数的贝叶斯分位回归模型

研究目标：建立零膨胀损失次数的贝叶斯分位回归模型。研究方法：通过增加随机扰动将离散型的损失次数数据转化为连续型数据,在预测误差平方和最小的条件下,求解出分位数水平,并应用贝叶斯方法求解分位回归模型中的参数。研究发现：基于得到的分位回归模型及相应的分位数水平,实现对未来的损失频率的预测。研究创新：借助等式关系,求解分位回归的分位数水平,避免主观选择分位数水平的弊端,实现对零膨胀损失次数贝叶斯分位回归建模。研究价值：基于一组实际数据的实证分析结果表明,该模型可以显著改进现有模型的拟合效果。