Canonical Correlation Analysis Through Linear Modeling

In this paper, we introduce linear modeling of canonical correlation analysis, which estimates canonical direction matrices by minimising a quadratic objective function. The linear modeling results in a class of estimators of canonical direction matrices, and an optimal class is derived in the sense described herein. The optimal class guarantees several of the following desirable advantages: first, its estimates of canonical direction matrices are asymptotically efficient; second, its test statistic for determining the number of canonical covariates always has a chi‐squared distribution asymptotically; third, it is straight forward to construct tests for variable selection. The standard canonical correlation analysis and other existing methods turn out to be suboptimal members of the class. Finally, we study the role of canonical variates as a means of dimension reduction for predictors and responses in multivariate regression. Numerical studies and data analysis are presented.     

多图模型及其在宏观经济指标相关分析中的应用

多图模型表示来自于不同类的同一组随机变量间的相关关系,结点表示随机变量,边表示变量之间的直接联系,各类的图模型反映了各自相关结构特征和类间共同的信息。用多图模型联合估计方法,将来自不同个体的数据按其特征分类,假设每类中各变量间的相依结构服从同一个高斯图模型,应用组Lasso方法和图Lasso方法联合估计每类的图模型结构。数值模拟验证了多图模型联合估计方法的有效性。用多图模型和联合估计方法对中国15个省份13个宏观经济指标进行相依结构分析,结果表明,不同经济发展水平省份的宏观经济变量间存在共同的相关联系,反映了中国现阶段经济发展的特征;每一类的相关结构反映了各类省份经济发展独有的特征。     

A Note on Multiple Regression for Single Index Model

Abstract

A simple method based on sliced inverse regression (SIR) is proposed to explore an effective dimension reduction (EDR) vector for the single index model. We avoid the principle component analysis step of the original SIR by using two sample mean vectors in two slices of the response variable and their difference vector. The theories become simpler, the method is equivalent to the multiple linear regression with dichotomized response, and the estimator can be expressed by a closed form, although the objective function might be an unknown nonlinear. It can be applied for the case when the number of covariates is large, and it requires no matrix operation or iterative calculation.     

Testing Serial Correlation in Single Index Models

In this article, we propose two testing procedures for the serial correlation in single index models by virtue of B spline approximation for unknown single index function. Under some regular conditions, we show that our proposed statistics asymptotically follow normal and χ² distribution. Many numerical studies illustrate that the proposed procedures can perform very well for moderate sample size.     

Linear Models Methods for Some Rank Function Analyses of Ordinal Categorical Data

This paper is concerned with the analysis of ordinal data through linear models for rank function measures.Primary attention is directed at pairwise Mann-Whitney statistics for which dimension reduction is managed by use of a Bradley-Terry log-linear structure.The nature of linear models for such quantities is contrasted with that for mean ranks (or ridits).Aspects of application are illustrated with an example for which results of other methods are also given.     

Locally efficient semiparametric estimators for a class of Poisson models with measurement error

The presence of measurement error may cause bias in parameter estimation and can lead to incorrect conclusions in data analyses. Despite a large body of literature on general measurement error problems, relatively few works exist to handle Poisson models. In this article we thoroughly study Poisson models with errors in covariates and propose consistent and locally efficient semiparametric estimators. We assess the finite sample performance of the estimators through extensive simulation studies and illustrate the proposed methodologies by analyzing data from the Stroke Recovery in Underserved Populations Study. The Canadian Journal of Statistics 47: 157–181; 2019 © 2019 Statistical Society of Canada     

Score estimation in the monotone single‐index model

We consider estimation in the single‐index model where the link function is monotone. For this model, a profile least‐squares estimator has been proposed to estimate the unknown link function and index. Although it is natural to propose this procedure, it is still unknown whether it produces index estimates that converge at the parametric rate. We show that this holds if we solve a score equation corresponding to this least‐squares problem. Using a Lagrangian formulation, we show how one can solve this score equation without any reparametrization. This makes it easy to solve the score equations in high dimensions. We also compare our method with the effective dimension reduction and the penalized least‐squares estimator methods, both available on CRAN as R packages, and compare with link‐free methods, where the covariates are elliptically symmetric.     

A Class of Semiparametric Transformation Models for Doubly Censored Failure Time Data

Doubly censored failure time data occur in many areas including demographical studies, epidemiology studies, medical studies and tumorigenicity experiments, and correspondingly some inference procedures have been developed in the literature (Biometrika, 91, 2004, 277; Comput. Statist. Data Anal., 57, 2013, 41; J. Comput. Graph. Statist., 13, 2004, 123). In this paper, we discuss regression analysis of such data under a class of flexible semiparametric transformation models, which includes some commonly used models for doubly censored data as special cases. For inference, the non‐parametric maximum likelihood estimation will be developed and in particular, we will present a novel expectation–maximization algorithm with the use of subject‐specific independent Poisson variables. In addition, the asymptotic properties of the proposed estimators are established and an extensive simulation study suggests that the proposed methodology works well for practical situations. The method is applied to an AIDS study.     

Estimation of value-at-risk using single index quantile regression

ABSTRACT

Value-at-Risk (VaR) is one of the best known and most heavily used measures of financial risk. In this paper, we introduce a non-iterative semiparametric model for VaR estimation called the single index quantile regression time series (SIQRTS) model. To test its performance, we give an application to four major US market indices: the S&P 500 Index, the Russell 2000 Index, the Dow Jones Industrial Average, and the NASDAQ Composite Index. Our results suggest that this method has a good finite sample performance and often outperforms a number of commonly used methods.     

Efficient estimation in partially linear single‐index models for longitudinal data

In this paper, we consider the estimation of both the parameters and the nonparametric link function in partially linear single‐index models for longitudinal data that may be unbalanced. In particular, a new three‐stage approach is proposed to estimate the nonparametric link function using marginal kernel regression and the parametric components with generalized estimating equations. The resulting estimators properly account for the within‐subject correlation. We show that the parameter estimators are asymptotically semiparametrically efficient. We also show that the asymptotic variance of the link function estimator is minimized when the working error covariance matrices are correctly specified. The new estimators are more efficient than estimators in the existing literature. These asymptotic results are obtained without assuming normality. The finite‐sample performance of the proposed method is demonstrated by simulation studies. In addition, two real‐data examples are analyzed to illustrate the methodology.     

Estimation and Inference Procedures for Semiparametric Distribution Models with Varying Linear‐Index

More flexible semiparametric linear‐index regression models are proposed to describe the conditional distribution. Such a model formulation captures varying effects of covariates over the support of a response variable in distribution, offers an alternative perspective on dimension reduction and covers a lot of widely used parametric and semiparameteric regression models. A feasible pseudo likelihood approach, accompanied with a simple and easily implemented algorithm, is further developed for the mixed case with both varying and invariant coefficients. By justifying some theoretical properties on Banach spaces, the uniform consistency and asymptotic Gaussian process of the proposed estimator are also established in this article. In addition, under the monotonicity of distribution in linear‐index, we develop an alternative approach based on maximizing a varying accuracy measure. By virtue of the asymptotic recursion relation for the estimators, some of the achievements in this direction include showing the convergence of the iterative computation procedure and establishing the large sample properties of the resulting estimator. It is noticeable that our theoretical framework is very helpful in constructing confidence bands for the parameters of interest and tests for the hypotheses of various qualitative structures in distribution. Generally, the developed estimation and inference procedures perform quite satisfactorily in the conducted simulations and are demonstrated to be useful in reanalysing data from the Boston house price study and the World Values Survey.     

Efficient Estimation in Marginal Partially Linear Models for Longitudinal/Clustered Data Using Splines

Abstract.  We consider marginal semiparametric partially linear models for longitudinal/clustered data and propose an estimation procedure based on a spline approximation of the non-parametric part of the model and an extension of the parametric marginal generalized estimating equations (GEE). Our estimates of both parametric part and non-parametric part of the model have properties parallel to those of parametric GEE, that is, the estimates are efficient if the covariance structure is correctly specified and they are still consistent and asymptotically normal even if the covariance structure is misspecified. By showing that our estimate achieves the semiparametric information bound, we actually establish the efficiency of estimating the parametric part of the model in a stronger sense than what is typically considered for GEE. The semiparametric efficiency of our estimate is obtained by assuming only conditional moment restrictions instead of the strict multivariate Gaussian error assumption.     

Instrument Assisted Regression for Errors in Variables Models with Binary Response

We study errors‐in‐variables problems when the response is binary and instrumental variables are available. We construct consistent estimators through taking advantage of the prediction relation between the unobservable variables and the instruments. The asymptotic properties of the new estimator are established and illustrated through simulation studies. We also demonstrate that the method can be readily generalized to generalized linear models and beyond. The usefulness of the method is illustrated through a real data example.     

Influence Functions for Sliced Inverse Regression

Abstract.  Sliced inverse regression (SIR) is a dimension reduction technique that is both efficient and simple to implement. The procedure itself relies heavily on estimates that are known to be highly non-robust and, as such, the issue of robustness is often raised. This paper looks at the robustness of SIR by deriving and plotting the influence function for a variety of contamination structures. The sample influence function is also considered and used to highlight that common outlier detection and deletion methods may not be entirely useful to SIR. The asymptotic variance of the estimates is also derived for the single index model when the explanatory variable is known to be normally distributed. The asymptotic variance is then compared for varying choices of the number of slices for a simple model example.     

典型相关分析的延拓研究

在典型相关分析中,求得典型相关变量的表达式并没有全部完成任务,例如需要确定典型相关变量的个数和变量选择。针对典型相关变量的个数问题,发现了常用的卡方检验和冗余分析方法的不足,进而提出了一种新的算法。针对原始变量的选择问题,提出了三种可能的路径。最后利用人体尺寸数据对相关结论进行了验证。     

A class of partially linear single‐index survival models

The authors define a class of “partially linear single‐index” survival models that are more flexible than the classical proportional hazards regression models in their treatment of covariates. The latter enter the proposed model either via a parametric linear form or a nonparametric single‐index form. It is then possible to model both linear and functional effects of covariates on the logarithm of the hazard function and if necessary, to reduce the dimensionality of multiple covariates via the single‐index component. The partially linear hazards model and the single‐index hazards model are special cases of the proposed model. The authors develop a likelihood‐based inference to estimate the model components via an iterative algorithm. They establish an asymptotic distribution theory for the proposed estimators, examine their finite‐sample behaviour through simulation, and use a set of real data to illustrate their approach.     

A Compressive Sensing Based Analysis of Anomalies in Generalized Linear Models

In this article, we present a compressive sensing based framework for generalized linear model regression that employs a two-component noise model and convex optimization techniques to simultaneously detect outliers and determine optimally sparse representations of noisy data from arbitrary sets of basis functions. We then extend our model to include model order reduction capabilities that can uncover inherent sparsity in regression coefficients and achieve simple, superior fits. Second, we use the mixed ?₂/?₁ norm to develop another model that can efficiently uncover block-sparsity in regression coefficients. By performing model order reduction over all independent variables and basis functions, our algorithms successfully deemphasize the effect of independent variables that become uncorrelated with dependent variables. This desirable property has various applications in real-time anomaly detection, such as faulty sensor detection and sensor jamming in wireless sensor networks. After developing our framework and inheriting a stable recovery theorem from compressive sensing theory, we present two simulation studies on sparse or block-sparse problems that demonstrate the superior performance of our algorithms with respect to (1) classic outlier-invariant regression techniques like least absolute value and iteratively reweighted least-squares and (2) classic sparse-regularized regression techniques like LASSO.     

股指时问序列的分形插值模型与R／S分析

运用分形插值模型和R／s分析法研究股指时间序列的变化规律和结构特征,通过建立分形插值模型刻画上证综合指数在一定时间内的变化规律,并预测其在短期内的指数走势。使用R／S分析法和Hurst指数,分析了上证综指的结构特征,指出市场具有状态持续性和分形分布等统计特征。     

基于GARCH族模型的中国旅游酒店板块指数收益率波动分析

运用GARCH族模型分析旅游酒店板块指数日收益率的波动特征,研究表明：旅游酒店板块收益率是一个平稳过程,其波动具有“聚集”现象和“非对称效应”。GARCH（2,1）模型比GARCH（1,1）模型更好地消除了收益率序列的异方差性;TARCH（2,1）模型的拟合效果最好;GARCH—M模型和非对称的CARCH（1,1）模型都不适用于描述收益率的波动特征。