Bayesian composite quantile regression for linear mixed-effects models

Longitudinal data are commonly modeled with the normal mixed-effects models. Most modeling methods are based on traditional mean regression, which results in non robust estimation when suffering extreme values or outliers. Median regression is also not a best choice to estimation especially for non normal errors. Compared to conventional modeling methods, composite quantile regression can provide robust estimation results even for non normal errors. In this paper, based on a so-called pseudo composite asymmetric Laplace distribution (PCALD), we develop a Bayesian treatment to composite quantile regression for mixed-effects models. Furthermore, with the location-scale mixture representation of the PCALD, we establish a Bayesian hierarchical model and achieve the posterior inference of all unknown parameters and latent variables using Markov Chain Monte Carlo (MCMC) method. Finally, this newly developed procedure is illustrated by some Monte Carlo simulations and a case analysis of HIV/AIDS clinical data set.     

Nonstationary nonlinear quantile regression

This study examines estimation and inference based on quantile regression for parametric nonlinear models with an integrated time series covariate. We first derive the limiting distribution of the nonlinear quantile regression estimator and then consider testing for parameter restrictions, when the regression function is specified as an asymptotically homogeneous function. We also study linear-in-parameter regression models when the regression function is given by integrable regression functions as well as asymptotically homogeneous regression functions. We, furthermore, propose a fully modified estimator to reduce the bias in the original estimator under a certain set of conditions. Finally, simulation studies show that the estimators behave well, especially when the regression error term has a fat-tailed distribution.     

Mixtures of Linear Regression with Measurement Errors

Existing research on mixtures of regression models are limited to directly observed predictors. The estimation of mixtures of regression for measurement error data imposes challenges for statisticians. For linear regression models with measurement error data, the naive ordinary least squares method, which directly substitutes the observed surrogates for the unobserved error-prone variables, yields an inconsistent estimate for the regression coefficients. The same inconsistency also happens to the naive mixtures of regression estimate, which is based on the traditional maximum likelihood estimator and simply ignores the measurement error. To solve this inconsistency, we propose to use the deconvolution method to estimate the mixture likelihood of the observed surrogates. Then our proposed estimate is found by maximizing the estimated mixture likelihood. In addition, a generalized EM algorithm is also developed to find the estimate. The simulation results demonstrate that the proposed estimation procedures work well and perform much better than the naive estimates.     

Quantile Regression for Location‐Scale Time Series Models with Conditional Heteroscedasticity

This paper considers quantile regression for a wide class of time series models including autoregressive and moving average (ARMA) models with asymmetric generalized autoregressive conditional heteroscedasticity errors. The classical mean‐variance models are reinterpreted as conditional location‐scale models so that the quantile regression method can be naturally geared into the considered models. The consistency and asymptotic normality of the quantile regression estimator is established in location‐scale time series models under mild conditions. In the application of this result to ARMA‐generalized autoregressive conditional heteroscedasticity models, more primitive conditions are deduced to obtain the asymptotic properties. For illustration, a simulation study and a real data analysis are provided.     

Collaborative sliced inverse regression

Sliced inverse regression (SIR) is an effective method for dimensionality reduction in high-dimensional regression problems. However, the method has requirements on the distribution of the predictors that are hard to check since they depend on unobserved variables. It has been shown that, if the distribution of the predictors is elliptical, then these requirements are satisfied. In case of mixture models, the ellipticity is violated and in addition there is no assurance of a single underlying regression model among the different components. Our approach clusterizes the predictors space to force the condition to hold on each cluster and includes a merging technique to look for different underlying models in the data. A study on simulated data as well as two real applications are provided. It appears that SIR, unsurprisingly, is not capable of dealing with a mixture of Gaussians involving different underlying models whereas our approach is able to correctly investigate the mixture.     

空间回归模型估计中的最小二乘法

空间回归模型由于引入了空间地理信息而使得其参数估计变得复杂,因为主要采用最大似然法,致使一般人认为在空间回归模型参数估计中不存在最小二乘法。通过分析空间回归模型的参数估计技术,研究发现,最小二乘法和最大似然法分别用于估计空间回归模型的不同的参数,只有将两者结合起来才能快速有效地完成全部的参数估计。数理论证结果表明,空间回归模型参数最小二乘估计量是最佳线性无偏估计量。空间回归模型的回归参数可以在估计量为正态性的条件下而实施显著性检验,而空间效应参数则不可以用此方法进行检验。     

Linearized Restricted Ridge Regression Estimator in Linear Regression

This article primarily aims to put forward the linearized restricted ridge regression (LRRR) estimator in linear regression models. Two types of LRRR estimators are investigated under the PRESS criterion and the optimal LRRR estimators and the optimal restricted generalized ridge regression estimator are obtained. We apply the results to the Hald data and finally make a simulation study by using the method of McDonald and Galarneau.     

Regression Tolerance Intervals

In this article, we discuss the utility of tolerance intervals for various regression models. We begin with a discussion of tolerance intervals for linear and nonlinear regression models. We then introduce a novel method for constructing nonparametric regression tolerance intervals by extending the well-established procedure for univariate data. Simulation results and application to real datasets are presented to help visualize regression tolerance intervals and to demonstrate that the methods we discuss have coverage probabilities very close to the specified nominal confidence level.     

A robust and efficient estimation method for nonparametric models with jump points

Nonparametric models with jump points have been considered by many researchers. However, most existing methods based on least squares or likelihood are sensitive when there are outliers or the error distribution is heavy tailed. In this article, a local piecewise-modal method is proposed to estimate the regression function with jump points in nonparametric models, and a piecewise-modal EM algorithm is introduced to estimate the proposed estimator. Under some regular conditions, the large-sample theory is established for the proposed estimators. Several simulations are presented to evaluate the performances of the proposed method, which shows that the proposed estimator is more efficient than the local piecewise-polynomial regression estimator in the presence of outliers or heavy tail error distribution. What is more, the proposed procedure is asymptotically equivalent to the local piecewise-polynomial regression estimator under the assumption that the error distribution is a Gaussian distribution. The proposed method is further illustrated via the sea-level pressures.     

Estimation of the Variance Function in Heteroscedastic Linear Regression Models

Heteroscedasticity generally exists when a linear regression model is applied to analyzing some real-world problems. Therefore, how to accurately estimate the variance functions of the error term in a heteroscedastic linear regression model is of great importance for obtaining efficient estimates of the regression parameters and making valid statistical inferences. A method for estimating the variance function of heteroscedastic linear regression models is proposed in this article based on the variance-reduced local linear smoothing technique. Some simulations and comparisons with other method are conducted to assess the performance of the proposed method. The results demonstrate that the proposed method can accurately estimate the variance functions and therefore produce more efficient estimates of the regression parameters.     

Dynamic approach to linear statistical calibration with an application in microwave radiometry

The problem of statistical calibration of a measuring instrument can be framed both in a statistical context as well as in an engineering context. In the first, the problem is dealt with by distinguishing between the ‘classical’ approach and the ‘inverse’ regression approach. Both of these models are static models and are used to estimate exact measurements from measurements that are affected by error. In the engineering context, the variables of interest are considered to be taken at the time at which you observe it. The Bayesian time series analysis method of Dynamic Linear Models can be used to monitor the evolution of the measures, thus introducing a dynamic approach to statistical calibration. The research presented employs a new approach to performing statistical calibration. A simulation study in the context of microwave radiometry is conducted that compares the dynamic model to traditional static frequentist and Bayesian approaches. The focus of the study is to understand how well the dynamic statistical calibration method performs under various signal-to-noise ratios, r.     

Saddlepoint-Based Bootstrap Inference for Quadratic Estimating Equations

Abstract.  We propose an easy to implement method for making small sample parametric inference about the root of an estimating equation expressible as a quadratic form in normal random variables. It is based on saddlepoint approximations to the distribution of the estimating equation whose unique root is a parameter's maximum likelihood estimator (MLE), while substituting conditional MLEs for the remaining (nuisance) parameters. Monotoncity of the estimating equation in its parameter argument enables us to relate these approximations to those for the estimator of interest. The proposed method is equivalent to a parametric bootstrap percentile approach where Monte Carlo simulation is replaced by saddlepoint approximation. It finds applications in many areas of statistics including, nonlinear regression, time series analysis, inference on ratios of regression parameters in linear models and calibration. We demonstrate the method in the context of some classical examples from nonlinear regression models and ratios of regression parameter problems. Simulation results for these show that the proposed method, apart from being generally easier to implement, yields confidence intervals with lengths and coverage probabilities that compare favourably with those obtained from several competing methods proposed in the literature over the past half-century.     

Semiparametric estimation for seasonal long-memory time series using generalized exponential models

We consider a generalized exponential (GEXP) model in the frequency domain for modeling seasonal long-memory time series. This model generalizes the fractional exponential (FEXP) model [Beran, J., 1993. Fitting long-memory models by generalized linear regression. Biometrika 80, 817–822] to allow the singularity in the spectral density occurring at an arbitrary frequency for modeling persistent seasonality and business cycles. Moreover, the short-memory structure of this model is characterized by the Bloomfield [1973. An exponential model for the spectrum of a scalar time series. Biometrika 60, 217–226] model, which has a fairly flexible semiparametric form. The proposed model includes fractionally integrated processes, Bloomfield models, FEXP models as well as GARMA models [Gray, H.L., Zhang, N.-F., Woodward, W.A., 1989. On generalized fractional processes. J. Time Ser. Anal. 10, 233–257] as special cases. We develop a simple regression method for estimating the seasonal long-memory parameter. The asymptotic bias and variance of the corresponding long-memory estimator are derived. Our methodology is applied to a sunspot data set and an Internet traffic data set for illustration.     

Application of Quasi-Least Squares to Analyse Replicated Autoregressive Time Series Regression Models

Time series regression models have been widely studied in the literature by several authors. However, statistical analysis of replicated time series regression models has received little attention. In this paper, we study the application of the quasi-least squares method to estimate the parameters in a replicated time series model with errors that follow an autoregressive process of order p. We also discuss two other established methods for estimating the parameters: maximum likelihood assuming normality and the Yule-Walker method. When the number of repeated measurements is bounded and the number of replications n goes to infinity, the regression and the autocorrelation parameters are consistent and asymptotically normal for all three methods of estimation. Basically, the three methods estimate the regression parameter efficiently and differ in how they estimate the autocorrelation. When p=2, for normal data we use simulations to show that the quasi-least squares estimate of the autocorrelation is undoubtedly better than the Yule-Walker estimate. And the former estimate is as good as the maximum likelihood estimate almost over the entire parameter space.     

A model of regression lines through a common point: estimation of the focal point in wind-blown sand phenomena

This paper discusses a model in which the regression lines will be passing through a common point. This point exists as a focal point in the wind-blown sand phenomena. The model of regression lines will be called ‘the focal point regression model’. The focal point will move according to the conditions of the experiments or the measurement site, so it must be estimated together with regression coefficients. The existence of the focal point is mathematically proved in the research field of coastal engineering, but its physical meaning and exact estimation method have not been established. Considering the experimental and/or measurement conditions, five models, that is, common or different error variance(s), passing through or not the centroid and Bayes-like approach are proposed. Moreover, the formulae of direct computation for a focal point under some conditions are given for engineering purpose. The models are applied to the wind-blown sand data, and behaviors of the models are verified by numerical experiments.     

Regression Methods for Combining Multiple Classifiers

As no single classification method outperforms other classification methods under all circumstances, decision-makers may solve a classification problem using several classification methods and examine their performance for classification purposes in the learning set. Based on this performance, better classification methods might be adopted and poor methods might be avoided. However, which single classification method is the best to predict the classification of new observations is still not clear, especially when some methods offer similar classification performance in the learning set. In this article we present various regression and classical methods, which combine several classification methods to predict the classification of new observations. The quality of the combined classifiers is examined on some real data. Nonparametric regression is the best method of combining classifiers.     

Variance estimation in nonparametric regression with jump discontinuities

Variance estimation is an important topic in nonparametric regression. In this paper, we propose a pairwise regression method for estimating the residual variance. Specifically, we regress the squared difference between observations on the squared distance between design points, and then estimate the residual variance as the intercept. Unlike most existing difference-based estimators that require a smooth regression function, our method applies to regression models with jump discontinuities. Our method also applies to the situations where the design points are unequally spaced. Finally, we conduct extensive simulation studies to evaluate the finite-sample performance of the proposed method and compare it with some existing competitors.     

A consistent test for heteroscedasticity in semi-parametric regression with nonparametric variance function based on the kernel method

It is important to detect the variance heterogeneity in regression models. Heteroscedasticity tests have been well studied in parametric and nonparametric regression models. This paper presents a consistent test for heteroscedasticity for nonlinear semi-parametric regression models with nonparametric variance function based on the kernel method. The properties of the test are investigated through Monte Carlo simulations. The test methods are illustrated with a real example.     

Buckley–James-type estimation of quantile regression with recurrent gap time data

In longitudinal studies, an individual may potentially undergo a series of repeated recurrence events. The gap times, which are referred to as the times between successive recurrent events, are typically the outcome variables of interest. Various regression models have been developed in order to evaluate covariate effects on gap times based on recurrence event data. The proportional hazards model, additive hazards model, and the accelerated failure time model are all notable examples. Quantile regression is a useful alternative to the aforementioned models for survival analysis since it can provide great flexibility to assess covariate effects on the entire distribution of the gap time. In order to analyze recurrence gap time data, we must overcome the problem of the last gap time subjected to induced dependent censoring, when numbers of recurrent events exceed one time. In this paper, we adopt the Buckley–James-type estimation method in order to construct a weighted estimation equation for regression coefficients under the quantile model, and develop an iterative procedure to obtain the estimates. We use extensive simulation studies to evaluate the finite-sample performance of the proposed estimator. Finally, analysis of bladder cancer data is presented as an illustration of our proposed methodology.     

Heteroskedastic linear regression model with compositional response and covariates

Compositional data are known as a sort of complex multidimensional data with the feature that reflect the relative information rather than absolute information. There are a variety of models for regression analysis with compositional variables. Similar to the traditional regression analysis, the heteroskedasticity still exists in these models. However, the existing heteroskedastic regression analysis methods cannot apply in these models with compositional error term. In this paper, we mainly study the heteroskedastic linear regression model with compositional response and covariates. The parameter estimator is obtained through weighted least squares method. For the hypothesis test of parameter, the test statistic is based on the original least squares estimator and corresponding heteroskedasticity-consistent covariance matrix estimator. When the proposed method is applied to both simulation and real example, we use the original least squares method as a comparison during the whole process. The results implicate the model's practicality and effectiveness in regression analysis with heteroskedasticity.