Bayesian inference for bivariate generalized linear models in diagnosing renal arterial obstruction

Generalized linear models are well-established generalizations of the linear models used for regression and analysis of variance. They allow flexible mean structures and general distributions, other than the linear link and normal response assumed in regression. Further enhancements using ideas from multivariate analysis improve power and precision by modelling dependencies between response variables. This paper focuses on the specific case of regression models for bivariate Bernoulli responses and investigates their analysis using a Bayesian approach. The important problem of renal arterial obstruction is considered, as a medical application of these models.     

A New Regression Model: Modal Linear Regression

The mode of a distribution provides an important summary of data and is often estimated on the basis of some non‐parametric kernel density estimator. This article develops a new data analysis tool called modal linear regression in order to explore high‐dimensional data. Modal linear regression models the conditional mode of a response Y given a set of predictors x as a linear function of x . Modal linear regression differs from standard linear regression in that standard linear regression models the conditional mean (as opposed to mode) of Y as a linear function of x . We propose an expectation–maximization algorithm in order to estimate the regression coefficients of modal linear regression. We also provide asymptotic properties for the proposed estimator without the symmetric assumption of the error density. Our empirical studies with simulated data and real data demonstrate that the proposed modal regression gives shorter predictive intervals than mean linear regression, median linear regression and MM‐estimators.     

Phase I Analysis of Multiple Linear Regression Profiles

This article considers the Phase I analysis of data when the quality of a process or product is characterized by a multiple linear regression model. This is usually referred to as the analysis of linear profiles in the statistical quality control literature. The literature includes several approaches for the analysis of simple linear regression profiles. Little work, however, has been done in the analysis of multiple linear regression profiles. This article proposes a new approach for the analysis of Phase I multiple linear regression profiles. Using this approach, regardless of the number of explanatory variables used to describe it, the profile response is monitored using only three parameters, an intercept, a slope, and a variance. Using simulation, the performance of the proposed method is compared to that of the existing methods for monitoring multiple linear profiles data in terms of the probability of a signal. The advantage of the proposed method over the existing methods is greatly improved detection of changes in the process parameters of linear profiles with high-dimensional space. The article also proposes useful diagnostic aids based on F-statistics to help in identifying the source of profile variation and the locations of out-of-control samples. Finally, the use of multiple linear profile methods is illustrated by a data set from a calibration application at National Aeronautics and Space Administration (NASA) Langley Research Center.     

Empirical Likelihood Based Synthetic Data Method for Censored Regression Analysis

In this article, we propose a new empirical likelihood method for linear regression analysis with a right censored response variable. The method is based on the synthetic data approach for censored linear regression analysis. A log-empirical likelihood ratio test statistic for the entire regression coefficients vector is developed and we show that it converges to a standard chi-squared distribution. The proposed method can also be used to make inferences about linear combinations of the regression coefficients. Moreover, the proposed empirical likelihood ratio provides a way to combine different normal equations derived from various synthetic response variables. Maximizing this empirical likelihood ratio yields a maximum empirical likelihood estimator which is asymptotically equivalent to the solution of the estimating equation that are optimal linear combination of the original normal equations. It improves the estimation efficiency. The method is illustrated by some Monte Carlo simulation studies as well as a real example.     

Multiple deletion measures and conditional influence in regression model

The joint effect of the deletion of the ith and jih cases is given by Gray and Ling (1984), they discussed the influence measures for influential subsets in linear regression analysis. The present paper is concerned with multiple sets of deletion measures in the linear regression model. In particular we are interested in the effects of the jointly and conditional influence analysis for the detection of two influential subsets.     

Trimmed least squares estimator as best trimmed linear conditional estimator for linear regression model

A class of trimmed linear conditional estimators based on regression quantiles for the linear regression model is introduced. This class serves as a robust analogue of non-robust linear unbiased estimators. Asymptotic analysis then shows that the trimmed least squares estimator based on regression quantiles ( Koenker and Bassett ( 1978 ) ) is the best in this estimator class in terms of asymptotic covariance matrices. The class of trimmed linear conditional estimators contains the Mallows-type bounded influence trimmed means ( see De Jongh et al ( 1988 ) ) and trimmed instrumental variables estimators. A large sample methodology based on trimmed instrumental variables estimator for confidence ellipsoids and hypothesis testing is also provided.     

国内删失数据统计研究状况综述

研究了国内在线性回归模型、非线性回归模型、半参数回归、非参数回归、单指标回归、生存分析、时间序列分析、密度估计等领域删失数据统计研究状况。     

Latent root regression: a biased regression methodology for use with collinear predictor variables

Many different biased regression techniques have been proposed for estimating parameters of a multiple linear regression model when the predictor variables are collinear. One particular alternative, latent root regression analysis, is a technique based on analyzing the latent roots and latent vectors of the correlation matrix of both the response and the predictor variables. It is the purpose of this paper to review the latent root regression estimator and to re-examine some of its properties and applications. It is shown that the latent root estimator is a member of a wider class of estimators for linear models     

Chapter Notes

Tests for redundancy of variables in linear two-group discriminant analysis are well known and frequently used. We give a survey of similar tests, including the one-sample T ² as a special case, in the situation in which only the mean vector (but no covariance matrix) is available in one sample. Then we show that a relation between linear regression and discriminant functions found by Fisher (1936) can be generalized to this situation. Relating regression and discriminant analysis to a multivariate linear model sheds more light on the relationship between them. Practical and didactical advantages of the regression approach to T ² tests and discriminant analysis are outlined.     

A note on functional linear regression

This work focuses on the linear regression model with functional covariate and scalar response. We compare the performance of two (parametric) linear regression estimators and a nonparametric (kernel) estimator via a Monte Carlo simulation study and the analysis of two real data sets. The first linear estimator expands the predictor and the regression weight function in terms of the trigonometric basis, while the second one uses functional principal components. The choice of the regularization degree in the linear estimators is addressed.     

计量经济学和统计学视角下的线性回归模型——再议线性回归模型的设定

从属性、构建方法及意义等方面,分析研究线性回归模型在计量经济学和统计学两学科视角下的差异,并根据这种差异进一步提出回归模型的基本设定思路。研究表明:识别这种差异是完成模型设定工作的基础性和必要性举措,有助于实现线性回归模型的正确设定。以经典例证对计量经济学和统计学回归模型在应用中的区别以及模型设定问题进行进一步展示和分析。     

Estimating the structural dimension of regressions via parametric inverse regression

A new estimation method for the dimension of a regression at the outset of an analysis is proposed. A linear subspace spanned by projections of the regressor vector X , which contains part or all of the modelling information for the regression of a vector Y on X , and its dimension are estimated via the means of parametric inverse regression. Smooth parametric curves are fitted to the p inverse regressions via a multivariate linear model. No restrictions are placed on the distribution of the regressors. The estimate of the dimension of the regression is based on optimal estimation procedures. A simulation study shows the method to be more powerful than sliced inverse regression in some situations.     

广义线性混合模型在顾客满意度研究中的应用——基于某地区银行理财产品顾客满意度的分析

在对广义线性模型与经典线性模型进行对比分析基础上,重点介绍了广义线性混合模型与估计方法及其在满意度调查数据中的模型设定与应用,并采用某调查机构在2011年1月至2012年3月期间对购买过某地区银行理财产品的客户进行的满意度调查数据进行实证分析。研究表明:相对于经典线性回归模型与广义线性模型,广义线性混合模型是分析满意度调查数据的有效方法。     

Subset selection in quantile regression analysis via alternative Bayesian information criteria and heuristic optimization

Subset selection is an extensively studied problem in statistical learning. Especially it becomes popular for regression analysis. This problem has considerable attention for generalized linear models as well as other types of regression methods. Quantile regression is one of the most used types of regression method. In this article, we consider subset selection problem for quantile regression analysis with adopting some recent Bayesian information criteria. We also utilized heuristic optimization during selection process. Simulation and real data application results demonstrate the capability of the mentioned information criteria. According to results, these information criteria can determine the true models effectively in quantile regression models.     

Classical testing in functional linear models

We extend four tests common in classical regression – Wald, score, likelihood ratio and F tests – to functional linear regression, for testing the null hypothesis, that there is no association between a scalar response and a functional covariate. Using functional principal component analysis, we re-express the functional linear model as a standard linear model, where the effect of the functional covariate can be approximated by a finite linear combination of the functional principal component scores. In this setting, we consider application of the four traditional tests. The proposed testing procedures are investigated theoretically for densely observed functional covariates when the number of principal components diverges. Using the theoretical distribution of the tests under the alternative hypothesis, we develop a procedure for sample size calculation in the context of functional linear regression. The four tests are further compared numerically for both densely and sparsely observed noisy functional data in simulation experiments and using two real data applications.     

Linear trimmed means for the linear regression with AR(1) errors model

For the linear regression with AR(1) errors model, the robust generalized and feasible generalized estimators of Lai et al. (2003) of regression parameters are shown to have the desired property of a robust Gauss Markov theorem. This is done by showing that these two estimators are the best among classes of linear trimmed means. Monte Carlo and data analysis for this technique have been performed.     

LARS-type algorithm for group lasso

The least absolute shrinkage and selection operator (lasso) has been widely used in regression analysis. Based on the piecewise linear property of the solution path, least angle regression provides an efficient algorithm for computing the solution paths of lasso. Group lasso is an important generalization of lasso that can be applied to regression with grouped variables. However, the solution path of group lasso is not piecewise linear and hence cannot be obtained by least angle regression. By transforming the problem into a system of differential equations, we develop an algorithm for efficient computation of group lasso solution paths. Simulation studies are conducted for comparing the proposed algorithm to the best existing algorithm: the groupwise-majorization-descent algorithm.     

TWO-STAGE SUPPORT ESTIMATION

This paper presents a two‐stage procedure for estimating the conditional support curve of a random variable X, given the information of a random vector X. Quantile estimation is followed by an extremal analysis on the residuals for problems which can be written as regression models. The technique is applied to data from the National Bureau of Economic Research and US Census Bureau's Center for Economic Studies which contain all four‐digit manufacturing industries. Simulation results show that in linear regression models the proposed estimation procedure is more efficient than the extreme linear regression quantile.     

Covariance-regularized regression and classification for high dimensional problems

Summary.  We propose covariance-regularized regression, a family of methods for prediction in high dimensional settings that uses a shrunken estimate of the inverse covariance matrix of the features to achieve superior prediction. An estimate of the inverse covariance matrix is obtained by maximizing the log-likelihood of the data, under a multivariate normal model, subject to a penalty; it is then used to estimate coefficients for the regression of the response onto the features. We show that ridge regression, the lasso and the elastic net are special cases of covariance-regularized regression, and we demonstrate that certain previously unexplored forms of covariance-regularized regression can outperform existing methods in a range of situations. The covariance-regularized regression framework is extended to generalized linear models and linear discriminant analysis, and is used to analyse gene expression data sets with multiple class and survival outcomes.     

Partial Least Squares: A First-order Analysis

We compare the partial least squares (PLS) and the principal component analysis (PCA), in a general case in which the existence of a true linear regression is not assumed. We prove under mild conditions that PLS and PCA are equivalent, to within a first-order approximation, hence providing a theoretical explanation for empirical findings reported by other researchers. Next, we assume the existence of a true linear regression equation and obtain asymptotic formulas for the bias and variance of the PLS parameter estimator