Standardization of Variables and Collinearity Diagnostic in Ridge Regression

Ridge estimation (RE) is an alternative method to ordinary least squares when there exists a collinearity problem in a linear regression model. The variance inflator factor (VIF) is applied to test if the problem exists in the original model and is also necessary after applying the ridge estimate to check if the chosen value for parameter k has mitigated the collinearity problem. This paper shows that the application of the original data when working with the ridge estimate leads to non‐monotone VIF values. García et al. (2014) showed some problems with the traditional VIF used in RE. We propose an augmented VIF, VIF_R(j,k), associated with RE, which is obtained by standardizing the data before augmenting the model. The VIF_R(j,k) will coincide with the VIF associated with the ordinary least squares estimator when k = 0. The augmented VIF has the very desirable properties of being continuous, monotone in the ridge parameter and higher than one.     

Anomalies in the Foundations of Ridge Regression: Some Clarifications

Several anomalies in the foundations of ridge regression from the perspective of constrained least-square (LS) problems were pointed out in Jensen & Ramirez. Some of these so-called anomalies, attributed to the non-monotonic behaviour of the norm of unconstrained ridge estimators and the consequent lack of sufficiency of Lagrange's principle, are shown to be incorrect. It is noted in this paper that, for a fixed Y , norms of unconstrained ridge estimators corresponding to the given basis are indeed strictly monotone. Furthermore, the conditions for sufficiency of Lagrange's principle are valid for a suitable range of the constraint parameter. The discrepancy arose in the context of one data set due to confusion between estimates of the parameter vector, β , corresponding to different parametrization (choice of bases) and/or constraint norms. In order to avoid such confusion, it is suggested that the parameter β corresponding to each basis be labelled appropriately.     

Shrinkage Estimation Strategies in Generalised Ridge Regression Models: Low/High-Dimension Regime

In this study, we suggest pretest and shrinkage methods based on the generalised ridge regression estimation that is suitable for both multicollinear and high-dimensional problems. We review and develop theoretical results for some of the shrinkage estimators. The relative performance of the shrinkage estimators to some penalty methods is compared and assessed by both simulation and real-data analysis. We show that the suggested methods can be accounted as good competitors to regularisation techniques, by means of a mean squared error of estimation and prediction error. A thorough comparison of pretest and shrinkage estimators based on the maximum likelihood method to the penalty methods. In this paper, we extend the comparison outlined in his work using the least squares method for the generalised ridge regression.     

基于多元线性回归的上海市生产性服务业和汽车产业发展的关联分析

生产性服务业是上海市“十二五”重点发展产业,“十二五”期间是上海市汽车产业的关键时期,发展基于汽车行业的生产性服务业将有效促进上海市两大关键产业的发展.从上海市生产性服务业和汽车产业发展的实践出发,基于生产性服务业和汽车行业的关联性概念模型,应用多元线性回归技术,通过对上海市生产性服务业、汽车产业及其外部环境因素的多元线性回归,论文研究了上海市生产性服务业、汽车产业和外部环境的数量关联及其含义.     

企业与高校合作创新影响因素的回归分析

运用因子分析方法提取了影响合作创新的四个方面的因素,通过多元线性回归模型对企业高校合作创新的相关假设进行验证,并对检验结果进行了分析和探讨。     

W.F. Sheppard's Smoothing Method: A Precursor to Local Polynomial Regression

W.F. Sheppard has been much overlooked in the history of statistics although his work produced significant contributions. He developed a polynomial smoothing method and corrections of moment estimates for grouped data as well as extensive normal probability tables that have been widely used since the 20th century. Sheppard presented his smoothing method for actuaries in a series of publications during the early 20th century. Population data consist of irregularities, and some adjustment or smoothing of the data is often necessary. Simple techniques, such as Spencer's summation formulae involving arithmetic operations and moving averages, were commonly practised by actuaries to smooth out equally spaced data. Sheppard's method, however, is a polynomial smoothing method based on central differences. We will show how Sheppard's smoothing method was a significant milestone in the development of smoothing techniques and a precursor to local polynomial regression.     

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.     

A Selective Overview and Comparison of Robust Mixture Regression Estimators

Mixture regression models have been widely used in business, marketing and social sciences to model mixed regression relationships arising from a clustered and thus heterogeneous population. The unknown mixture regression parameters are usually estimated by maximum likelihood estimators using the expectation–maximisation algorithm based on the normality assumption of component error density. However, it is well known that the normality-based maximum likelihood estimation is very sensitive to outliers or heavy-tailed error distributions. This paper aims to give a selective overview of the recently proposed robust mixture regression methods and compare their performance using simulation studies.     

A Review and Comparison of Bandwidth Selection Methods for Kernel Regression

Over the last decades, several methods for selecting the bandwidth have been introduced in kernel regression. They differ quite a bit, and although there already exist more selection methods than for any other regression smoother, one can still observe coming up new ones. Given the need of automatic data‐driven bandwidth selectors for applied statistics, this review is intended to explain and, above all, compare these methods. About 20 different selection methods have been revised, implemented and compared in an extensive simulation study.     

Varying Coefficient Regression Models: A Review and New Developments

Varying coefficient regression models are known to be very useful tools for analysing the relation between a response and a group of covariates. Their structure and interpretability are similar to those for the traditional linear regression model, but they are more flexible because of the infinite dimensionality of the corresponding parameter spaces. The aims of this paper are to give an overview on the existing methodological and theoretical developments for varying coefficient models and to discuss their extensions with some new developments. The new developments enable us to use different amount of smoothing for estimating different component functions in the models. They are for a flexible form of varying coefficient models that requires smoothing across different covariates' spaces and are based on the smooth backfitting technique that is admitted as a powerful technique for fitting structural regression models and is also known to free us from the curse of dimensionality.     

Variable Selection and Shrinkage: Comparison of Some Approaches

A common strategy within the framework of regression models is the selection of variables with possible predictive value, which are incorporated in the regression model. Two recently proposed methods, Breiman's Garotte (B reiman , 1995) and Tibshirani's Lasso (T ibshirani , 1996) try to combine variable selection and shrinkage. We compare these with pure variable selection and shrinkage procedures. We consider the backward elimination procedure as a typical variable selection procedure and as an example of a shrinkage procedure an approach of V an H ouwelingen and L e C essie (1990). Additionally an extension of van Houwelingens and le Cessies approach proposed by S auerbrei (1999) is considered. The ordinary least squares method is used as a reference.
With the help of a simulation study we compare these approaches with respect to the distribution of the complexity of the selected model, the distribution of the shrinkage factors, selection bias, the bias and variance of the effect estimates and the average prediction error.     

SPSS相关分析与线性回归分析在我国物流业增加值分析中的应用

物流业增加值是反映物流业发展状况的核心指标,是衡量相关企业经营业绩及经济发展水平的重要依据．图而对其进行分析并在此基础上得出的结论具有重要的现实意义。文章以中国物流与采购网公布的有关统计数据为依托．通过SPSS相关分析及线性回归分析,对反映物流业发展状况的物流业增加值、物流固定资产投资、社会物流费用等指标的相互关系进行探讨．得出了一些对于物流业增加值的分析及预测有较强实际意义的结论,可供有关部门参考。     

A Survey of Differentially Private Regression for Clinical and Epidemiological Research

Differential privacy is a framework for data analysis that provides rigorous privacy protections for database participants. It has increasingly been accepted as the gold standard for privacy in the analytics industry, yet there are few techniques suitable for statistical inference in the health sciences. This is notably the case for regression, one of the most widely used modelling tools in clinical and epidemiological studies. This paper provides an overview of differential privacy and surveys the literature on differentially private regression, highlighting the techniques that hold the most relevance for statistical inference as practiced in clinical and epidemiological research. Research gaps and opportunities for further inquiry are identified.     

Nonuniform DIF Detection using Discriminant Logistic Analysis and Multinomial Logistic Regression: A comparison for polytomous items

This study focused on the effectiveness in nonuniform polytomous item DIF detection using Discriminant Logistic Analysis (DLA) and Multinomial Logistic Regression (MLR). A computer simulation study was conducted to compare the effect of using DLA and MLR, applying either an iterative test purification procedure or non-iterative to detect nonuniform DIF. The conditions under study were: DIF effect size (0.5, 1.0 and 1.5), sample size (500 and 1000), percentage of DIF items in the test (0, 10 and 20%) and DIF type (nonuniform). The results suggest that DLA is more accurate than MLR in detecting DIF. However, the purification process only improved the correct detection rate when MLR was applied. The false positive rates for both procedures were similar. Moreover, when the test purification procedure was used, the proportion of non-DIF items that were detected as DIF decreased for both procedures, although the false positive rates were smaller for DLA than for MLR.     

农户农地流转意愿影响因素的探讨——基于成都市实地调查及主成分改进Logistic回归分析

分析表明,影响农地流转意愿最大的三个因子依次为:非农工作所在地,非农工作人数比例以及农业人均年经营收入。结合成都统筹城乡改革的经验,本文建议采用规划民营工业园区、提高农业经营收入等方式,来促进农地的流转和整合,提高农民可支配收入,实现资源和收入的有效配置。     

Three Sides of Smoothing: Categorical Data Smoothing, Nonparametric Regression, and Density Estimation

The past forty years have seen a great deal of research into the construction and properties of nonparametric estimates of smooth functions. This research has focused primarily on two sides of the smoothing problem: nonparametric regression and density estimation. Theoretical results for these two situations are similar, and multivariate density estimation was an early justification for the Nadaraya-Watson kernel regression estimator.
A third, less well-explored, strand of applications of smoothing is to the estimation of probabilities in categorical data. In this paper the position of categorical data smoothing as a bridge between nonparametric regression and density estimation is explored. Nonparametric regression provides a paradigm for the construction of effective categorical smoothing estimates, and use of an appropriate likelihood function yields cell probability estimates with many desirable properties. Such estimates can be used to construct regression estimates when one or more of the categorical variables are viewed as response variables. They also lead naturally to the construction of well-behaved density estimates using local or penalized likelihood estimation, which can then be used in a regression context. Several real data sets are used to illustrate these points.     

HAC估计的性质与修正及其在伪回归中的应用研究

本文通过频谱分析揭示了预白化HAC和参数化VARHAC比非参数HAC具有明显优势,并指出传统预白化HAC法由于使用了存在有限样本偏差的OLS法来估计自回归参数,导致其存在有限样本偏差,由此构造的t统计量具有过度拒绝原假设倾向。为了减少预白化HAC的偏差,将OLS估计量的线性修正（LBC）和非线性修正（NBC）嵌入到预白化HAC中,研究表明该法能大大减少长期方差的估计偏差,并通过蒙特卡洛模拟证实对预白化HAC的修正估计能有效减少平稳过程之间的伪回归概率,从而提高了回归模型统计推断的可靠性。     

On random fuzzy variables of second order and their application to linear statistical inference with fuzzy data

This paper summarizes some results on random fuzzy variables with existing expectation and variance, called random fuzzy variables of second order. Using the Frechét-principle and – via support functions – the embedding of convex fuzzy sets into a Banach space of functions it especially presents a unified view on expectation and variance of random fuzzy variables. These notions are applied in developing linear statistical inference with fuzzy data. Detailed investigations are presented concerning best linear unbiased estimation in linear regression models with fuzzy observations. Received: November 1999     

全新范式方法在传统语言学研究中的应用

在对乔姆斯基的转换生成语法进行全面概述的基础上,指认出其对传统语言学研究范式所进行的三大转换,即从描述到解释,从表层到深层以及从解构到建构,而这种转换的实质则在于从经验到先验。     

Stock market reactions to upside and downside volatility of Bitcoin: A quantile analysis

We investigate how sensitive developed and emerging equity markets are to volatility dynamics of Bitcoin during tranquil, bear, and bull market regimes. Intraday price fluctuations of Bitcoin are represented by three measures of realized volatility, viz. total variance, upside semivariance, and downside semivariance. Our empirical analysis relies on a quantile regression framework, after orthogonalizing raw returns with respect to an array of relevant global factors and accounting for structural shifts in the series. The results suggest that developed-market returns are positively related to the realized variance proxy across various market conditions, while emerging-market returns are positively (negatively) correlated with realized variance during bear (normal and bull) market periods. The upside (downside) component of realized variance has a negative (positive) influence on returns of either market category, and the dependence structure is highly asymmetric across the return distribution. Additionally, we document that developed and emerging markets are more sensitive to downside volatility than to upside volatility when they enter tranquil or bull territory. Our results offer practical implications for policymakers and investors.