Infinite Parameter Estimates in Logistic Regression, with Application to Approximate Conditional Inference

This paper discusses recovery of information regarding logistic regression parameters in cases when maximum likelihood estimates of some parameters are infinite. An algorithm for detecting such cases and characterizing the divergence of the parameter estimates is presented. A method for fitting the remaining parameters is also presented . All of these methods rely only on sufficient statistics rather than less aggregated quantities, as required for inference according to the method of Kolassa & Tanner (1994). These results are applied to approximate conditional inference via saddlepoint methods. Specifically, the double saddlepoint method of Skovgaard (1987) is adapted to the case when the solution to the saddlepoint equations exists as a point at infinity     

Gaussian Regularized Sliced Inverse Regression

Sliced Inverse Regression (SIR) is an effective method for dimension reduction in high-dimensional regression problems. The original method, however, requires the inversion of the predictors covariance matrix. In case of collinearity between these predictors or small sample sizes compared to the dimension, the inversion is not possible and a regularization technique has to be used. Our approach is based on a Fisher Lecture given by R.D. Cook where it is shown that SIR axes can be interpreted as solutions of an inverse regression problem. We propose to introduce a Gaussian prior distribution on the unknown parameters of the inverse regression problem in order to regularize their estimation. We show that some existing SIR regularizations can enter our framework, which permits a global understanding of these methods. Three new priors are proposed leading to new regularizations of the SIR method. A comparison on simulated data as well as an application to the estimation of Mars surface physical properties from hyperspectral images are provided.     

Errors-in-Variables Regression Using Stein Estimates

A method is proposed for estimating regression parameters from data containing covariate measurement errors by using Stein estimates of the unobserved true covariates. The method produces consistent estimates for the slope parameter in the classical linear errors-in-variables model and applies to a broad range of nonlinear regression problems, provided the measurement error is Gaussian with known variance. Simulations are used to examine the performance of the estimates in a nonlinear regression problem and to compare them with the usual naive ones obtained by ignoring error and with other estimates proposed recently in the literature.     

Empirical Likelihood Based Rank Regression Inference

Rank regression procedures have been proposed and studied for numerous research applications that do not satisfy the underlying assumptions of the more common linear regression models. This article develops confidence regions for the slope parameter of rank regression using an empirical likelihood (EL) ratio method. It has the advantage of not requiring variance estimation which is required for the normal approximation method. The EL method is also range respecting and results in asymmetric confidence intervals. Simulation studies are used to compare and evaluate normal approximation versus EL inference methods for various conditions such as different sample size or error distribution. The simulation study demonstrates our proposed EL method almost outperforms the traditional method in terms of coverage probability, lower-tail side error, and upper-tail side error. An application of stability analysis also shows the EL method results in shorter confidence intervals for real life data.     

A Method for Bayesian Monotonic Multiple Regression

Abstract. When applicable, an assumed monotonicity property of the regression function w.r.t. covariates has a strong stabilizing effect on the estimates. Because of this, other parametric or structural assumptions may not be needed at all. Although monotonic regression in one dimension is well studied, the question remains whether one can find computationally feasible generalizations to multiple dimensions. Here, we propose a non‐parametric monotonic regression model for one or more covariates and a Bayesian estimation procedure. The monotonic construction is based on marked point processes, where the random point locations and the associated marks (function levels) together form piecewise constant realizations of the regression surfaces. The actual inference is based on model‐averaged results over the realizations. The monotonicity of the construction is enforced by partial ordering constraints, which allows it to asymptotically, with increasing density of support points, approximate the family of all monotonic bounded continuous functions.     

A Semiparametric Regression Method for Interval-Censored Data

In many medical studies, event times are recorded in an interval-censored (IC) format. For example, in numerous cancer trials, time to disease relapse is only known to have occurred between two consecutive clinic visits. Many existing modeling methods in the IC context are computationally intensive and usually require numerous assumptions that could be unrealistic or difficult to verify in practice. We propose a flexible and computationally efficient modeling strategy based on jackknife pseudo-observations (POs). The POs obtained based on nonparametric estimators of the survival function are employed as outcomes in an equivalent, yet simpler regression model that produces consistent covariate effect estimates. Hence, instead of operating in the IC context, the problem is translated into the realm of generalized linear models, where numerous options are available. Outcome transformations via appropriate link functions lead to familiar modeling contexts such as the proportional hazards and proportional odds. Moreover, the methods developed are not limited to these settings and have broader applicability. Simulations studies show that the proposed methods produce virtually unbiased covariate effect estimates, even for moderate sample sizes. An example from the International Breast Cancer Study Group (IBCSG) Trial VI further illustrates the practical advantages of this new approach.     

带测量误差的分位数回归模型的参数估计研究

近年来,研究预测变量带测量误差的分位数回归模型的参数估计问题已逐渐成为统计学中的一大热点问题,但由于带测量误差的分位数回归模型的参数估计十分复杂,所以相关研究很少,而目前的研究主要集中于误差分布为球形对称分布的正交回归法和误差分布自由的校正损失函数法等。在回归误差分布为正态分布的假设下,提出修正因子得分法(CFS),即在因子得分法的基础上进行参数估计,并对估计偏差进行修正得到最终估计;通过模拟研究,比较修正因子得分法相对正交回归估计(H-L估计)的优劣,并对修正因子得分法进行综合评价。     

Empirical Comparison of Nonparametric Regression Estimates on Real Data

The performance of nine different nonparametric regression estimates is empirically compared on ten different real datasets. The number of data points in the real datasets varies between 7, 900 and 18, 000, where each real dataset contains between 5 and 20 variables. The nonparametric regression estimates include kernel, partitioning, nearest neighbor, additive spline, neural network, penalized smoothing splines, local linear kernel, regression trees, and random forests estimates. The main result is a table containing the empirical L₂ risks of all nine nonparametric regression estimates on the evaluation part of the different datasets. The neural networks and random forests are the two estimates performing best. The datasets are publicly available, so that any new regression estimate can be easily compared with all nine estimates considered in this article by just applying it to the publicly available data and by computing its empirical L₂ risks on the evaluation part of the datasets.     

缺失偏态数据下线性回归模型的统计推断

研究缺失偏态数据下线性回归模型的参数估计问题,针对缺失偏态数据,为克服样本分布扭曲缺点和提高模型的回归系数、尺度参数和偏度参数的估计效果,提出了一种适合偏态数据下线性回归模型中缺失数据的修正回归插补方法.通过随机模拟和实例研究,并与均值插补、回归插补、随机回归插补方法比较,结果表明所提出的修正回归插补方法是有效可行的.     

高维混合效应模型的双正则化分位回归方法研究

针对高维混合效应模型,本文提出了一种双正则化分位回归方法.通过对随机和固定效应系数同时实施L1正则化惩罚,一方面能够对重要解释变量进行挑选,另一方面能够消除个体随机波动带来的偏差.求解参数估计的交替迭代算法不仅破解了要同时确定两个调整参数的难题,而且算法速度快.模拟结果也表明该方法不仅对误差类型有很强的抗干扰能力,同时在模型有不同稀疏程度时均表现良好,尤其是对于解释变量多于样本的高维情况.为了方便在实际问题中选择最优正则化参数,本文还对两种参数选取标准进行了比较研究.最后利用新方法对一个教育方面的数据进行了实证演示,找出了在各个分位点处对学生成绩有影响的重要因素.     

Robust Inference in Conditionally Linear Nonlinear Regression Models

Abstract.  We consider robust methods of likelihood and frequentist inference for the nonlinear parameter, say α , in conditionally linear nonlinear regression models. We derive closed-form expressions for robust conditional, marginal, profile and modified profile likelihood functions for α under elliptically contoured data distributions. Next, we develop robust exact-F confidence intervals for α and consider robust Fieller intervals for ratios of regression parameters in linear models. Several well-known examples are considered and Monte Carlo simulation results are presented.     

基于贝叶斯方法的比例数据分位数推断及其应用

为了尝试使用贝叶斯方法研究比例数据的分位数回归统计推断问题,首先基于Tobit模型给出了分位数回归建模方法,然后通过选取合适的先验分布得到了贝叶斯层次模型,进而给出了各参数的后验分布并用于Gibbs抽样。数值模拟分析验证了所提出的贝叶斯推断方法对于比例数据分析的有效性。最后,将贝叶斯方法应用于美国加州海洛因吸毒数据,在不同的分位数水平下揭示了吸毒频率的影响因素。     

非参数固定效应Panel Data模型的分位数回归推断

利用分位数回归方法,讨论了非参数固定效应Panel Data模型的估计和检验问题,得到了参数估计的渐近正态性及收敛速度。同时,建立一个秩得分(rank score)统计量来检验模型的固定效应,并证明了这个统计量渐近服从标准正态分布。     

Simulation-based Inference in a Zero-inflated Bernoulli Regression Model

The logistic regression model has become a standard tool to investigate the relationship between a binary outcome and a set of potential predictors. When analyzing binary data, it often arises that the observed proportion of zeros is greater than expected under the postulated logistic model. Zero-inflated binomial (ZIB) models have been developed to fit binary data that contain too many zeros. Maximum likelihood estimators in these models have been proposed and their asymptotic properties established. Several aspects of ZIB models still deserve attention however, such as the estimation of odds-ratios and event probabilities. In this article, we propose estimators of these quantities and we investigate their properties both theoretically and via simulations. Based on these results, we provide recommendations about the range of conditions (minimum sample size, maximum proportion of zeros in excess) under which a reliable statistical inference on the odds-ratios and event probabilities can be obtained in a ZIB regression model. A real-data example illustrates the proposed estimators.     

一种新的Boosting回归树方法

梯度Boosting思想在解释Boosting算法的运行机制时基于基学习器张成的空间为连续泛函空间,但是实际上在有限样本条件下形成的基学习器空间不一定是连续的。针对这一问题,从可加模型的角度出发,基于平方损失,提出一种重抽样提升回归树的新方法。该方法是一种加权的加法模型的逐步更新算法。实验结果表明,这种方法可以显著地提升一棵回归树的效果,减小预测误差,并且能得到比L2Boost算法更低的预测误差。     

Least Squares Consistent Estimates for Arbitrary Regression Functions Over an Abstract Space

In this article, we propose a new approach to sieve estimation for a general regression function when the dimension of the finite dimensional subspaces is a random quantity depending on the values of the observations.

The technique is introduced with the help of a simulation study on a functional linear model under extremely mild assumptions.

A sketch of the proof concerning the main statements is then given in the more general case when the regression function is not necessarily linear.     

Bayesian Inference in Multivariate Regression With Missing Observations on the Response Variables

We discuss the case of the multivariate linear model Y = XB + E with Y an (n × p) matrix, and so on, when there are missing observations in the Y matrix in a so-called nested pattern. We propose an analysis that arises by incorporating the predictive density of the missing observations in determining the posterior distribution of B, and its mean and variance matrix. This involves us with matric-T variables. The resulting analysis is illustrated with some Canadian economic data.     

A New Robust Regression Method Based on Particle Swarm Optimization

Regression analysis is one of methods widely used in prediction problems. Although there are many methods used for parameter estimation in regression analysis, ordinary least squares (OLS) technique is the most commonly used one among them. However, this technique is highly sensitive to outlier observation. Therefore, in literature, robust techniques are suggested when data set includes outlier observation. Besides, in prediction a problem, using the techniques that reduce the effectiveness of outlier and using the median as a target function rather than an error mean will be more successful in modeling these kinds of data. In this study, a new parameter estimation method using the median of absolute rate obtained by division of the difference between observation values and predicted values by the observation value and based on particle swarm optimization was proposed. The performance of the proposed method was evaluated with a simulation study by comparing it with OLS and some other robust methods in the literature.