Bayesian LASSO-Regularized quantile regression for linear regression models with autoregressive errors

Quantile regression (QR) is a natural alternative for depicting the impact of covariates on the conditional distributions of a outcome variable instead of the mean. In this paper, we investigate Bayesian regularized QR for the linear models with autoregressive errors. LASSO-penalized type priors are forced on regression coefficients and autoregressive parameters of the model. Gibbs sampler algorithm is employed to draw the full posterior distributions of unknown parameters. Finally, the proposed procedures are illustrated by some simulation studies and applied to a real data analysis of the electricity consumption.     

Adaptive order selection for autoregressive models

Autoregressive model is a popular method for analysing the time dependent data, where selection of order parameter is imperative. Two commonly used selection criteria are the Akaike information criterion (AIC) and the Bayesian information criterion (BIC), which are known to suffer the potential problems regarding overfit and underfit, respectively. To our knowledge, there does not exist a criterion in the literature that can satisfactorily perform under various situations. Therefore, in this paper, we focus on forecasting the future values of an observed time series and propose an adaptive idea to combine the advantages of AIC and BIC but to mitigate their weaknesses based on the concept of generalized degrees of freedom. Instead of applying a fixed criterion to select the order parameter, we propose an approximately unbiased estimator of mean squared prediction errors based on a data perturbation technique for fairly comparing between AIC and BIC. Then use the selected criterion to determine the final order parameter. Some numerical experiments are performed to show the superiority of the proposed method and a real data set of the retail price index of China from 1952 to 2008 is also applied for illustration.     

Identification of non-linear additive autoregressive models

Summary.  We propose a lag selection method for non-linear additive autoregressive models that is based on spline estimation and the Bayes information criterion. The additive structure of the autoregression function is used to overcome the 'curse of dimensionality', whereas the spline estimators effectively take into account such a structure in estimation. A stepwise procedure is suggested to implement the method proposed. A comprehensive Monte Carlo study demonstrates good performance of the method proposed and a substantial computational advantage over existing local-polynomial-based methods. Consistency of the lag selection method based on the Bayes information criterion is established under the assumption that the observations are from a stochastic process that is strictly stationary and strongly mixing, which provides the first theoretical result of this kind for spline smoothing of weakly dependent data.     

A class of finite mixture of quantile regressions with its applications

Mixture of linear regression models provide a popular treatment for modeling nonlinear regression relationship. The traditional estimation of mixture of regression models is based on Gaussian error assumption. It is well known that such assumption is sensitive to outliers and extreme values. To overcome this issue, a new class of finite mixture of quantile regressions (FMQR) is proposed in this article. Compared with the existing Gaussian mixture regression models, the proposed FMQR model can provide a complete specification on the conditional distribution of response variable for each component. From the likelihood point of view, the FMQR model is equivalent to the finite mixture of regression models based on errors following asymmetric Laplace distribution (ALD), which can be regarded as an extension to the traditional mixture of regression models with normal error terms. An EM algorithm is proposed to obtain the parameter estimates of the FMQR model by combining a hierarchical representation of the ALD. Finally, the iterated weighted least square estimation for each mixture component of the FMQR model is derived. Simulation studies are conducted to illustrate the finite sample performance of the estimation procedure. Analysis of an aphid data set is used to illustrate our methodologies.     

Influence diagnostics for Student-t censored linear regression models

In this paper, we extend the censored linear regression model with normal errors to Student-t errors. A simple EM-type algorithm for iteratively computing maximum-likelihood estimates of the parameters is presented. To examine the performance of the proposed model, case-deletion and local influence techniques are developed to show its robust aspect against outlying and influential observations. This is done by the analysis of the sensitivity of the EM estimates under some usual perturbation schemes in the model or data and by inspecting some proposed diagnostic graphics. The efficacy of the method is verified through the analysis of simulated data sets and modelling a real data set first analysed under normal errors. The proposed algorithm and methods are implemented in the R package CensRegMod.     

Detection of a change-point in student-t linear regression models

In this work the Schwarz Inforamtion Criterion (SIC) is used in order to locate a change-point in linear regression models with independent errors distributed according to the Student-t distribution. The methodology is applied to data sets from the financial area.     

Identifying rational distributed lag models using a joint test applied to the corner table

Often a distributed lag response pattern can be usefully represented in rational polynomial form. When the impulse response function decays, the corner table may be useful for model identification if appropriate statistical tests may be done. One or more joint tests are called for since use of the corner table involves studying groups of its elements. We consider an asymptotic x2 statistic that permits joint tests. We report simulation results showing that the distribution of this statistic follows the x ² distribution, for certain sample sizes and degrees of freedom, well enough to be useful in practice. With two data sets we illustrate how this statistic can be a useful aid when using the corner table.     

Separated hypotheses testing for autoregressive models with non-negative residuals

Normal residual is one of the usual assumptions in autoregressive model but sometimes in practice we are faced with non-negative residuals. In this paper, we have derived modified maximum likelihood estimators of parameters of the residuals and autoregressive coefficient. Also asymptotic distribution of modified maximum likelihood estimators in both stationary and non-stationary models are computed. So that, we can derive asymptotic distribution of unit root, Vuong's and Cox's tests statistics in stationary situation. Using simulation, it shows that Akaike information criterion and Vuong's test work to select the optimal autoregressive model with non-negative residuals. Sometimes Vuong's test select two competing models as equivalent models. These models may be suitable or unsuitable equivalent models. So we consider Cox's test to make inference after model selection. Kolmogorov–Smirnov test confirms our results. Also we have computed tracking interval for competing models to choosing between two close competing models when Vuong's test and Cox's test cannot detect the differences.     

A lack-of-fit test for parametric zero-inflated Poisson models

Count data often contain many zeros. In parametric regression analysis of zero-inflated count data, the effect of a covariate of interest is typically modelled via a linear predictor. This approach imposes a restrictive, and potentially questionable, functional form on the relation between the independent and dependent variables. To address the noted restrictions, a flexible parametric procedure is employed to model the covariate effect as a linear combination of fixed-knot cubic basis splines or B-splines. The semiparametric zero-inflated Poisson regression model is fitted by maximizing the likelihood function through an expectation–maximization algorithm. The smooth estimate of the functional form of the covariate effect can enhance modelling flexibility. Within this modelling framework, a log-likelihood ratio test is used to assess the adequacy of the covariate function. Simulation results show that the proposed test has excellent power in detecting the lack of fit of a linear predictor. A real-life data set is used to illustrate the practicality of the methodology.     

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.     

Bayesian bridge-randomized penalized quantile regression estimation for linear regression model with AP(q) perturbation

Bridge penalized regression has many desirable statistical properties such as unbiasedness, sparseness as well as ‘oracle’. In Bayesian framework, bridge regularized penalty can be implemented based on generalized Gaussian distribution (GGD) prior. In this paper, we incorporate Bayesian bridge-randomized penalty and its adaptive version into the quantile regression (QR) models with autoregressive perturbations to conduct Bayesian penalization estimation. Employing the working likelihood of the asymmetric Laplace distribution (ALD) perturbations, the Bayesian joint hierarchical models are established. Based on the mixture representations of the ALD and generalized Gaussian distribution (GGD) priors of coefficients, the hybrid algorithms based on Gibbs sampler and Metropolis-Hasting sampler are provided to conduct fully Bayesian posterior estimation. Finally, the proposed Bayesian procedures are illustrated by some simulation examples and applied to a real data application of the electricity consumption.     

Unsupervised learning of regression mixture models with unknown number of components

ABSTRACT

We propose a new unsupervised learning algorithm to fit regression mixture models with unknown number of components. The developed approach consists in a penalized maximum likelihood estimation carried out by a robust expectation–maximization (EM)-like algorithm. We derive it for polynomial, spline, and B-spline regression mixtures. The proposed learning approach is unsupervised: (i) it simultaneously infers the model parameters and the optimal number of the regression mixture components from the data as the learning proceeds, rather than in a two-fold scheme as in standard model-based clustering using afterward model selection criteria, and (ii) it does not require accurate initialization unlike the standard EM for regression mixtures. The developed approach is applied to curve clustering problems. Numerical experiments on simulated and real data show that the proposed algorithm performs well and provides accurate clustering results, and confirm its benefit for practical applications.     

Focused information criterion and model averaging based on weighted composite quantile regression

We study the focused information criterion and frequentist model averaging and their application to post‐model‐selection inference for weighted composite quantile regression (WCQR) in the context of the additive partial linear models. With the non‐parametric functions approximated by polynomial splines, we show that, under certain conditions, the asymptotic distribution of the frequentist model averaging WCQR‐estimator of a focused parameter is a non‐linear mixture of normal distributions. This asymptotic distribution is used to construct confidence intervals that achieve the nominal coverage probability. With properly chosen weights, the focused information criterion based WCQR estimators are not only robust to outliers and non‐normal residuals but also can achieve efficiency close to the maximum likelihood estimator, without assuming the true error distribution. Simulation studies and a real data analysis are used to illustrate the effectiveness of the proposed procedure.     

Label switching and its solutions for frequentist mixture models

In this article, the label switching problem and the importance of solving it are discussed for frequentist mixture models if a simulation study is used to evaluate the performance of mixture model estimators. Two effective labelling methods are proposed by using true label for each observation. The empirical studies demonstrate that the new proposed methods work well and provide better results than the rule of thumb method of order constraint labelling. In addition, a Monte Carlo study also demonstrates that simple order constraint labelling can sometimes produce severely biased, and possibly meaningless, estimated bias and standard errors.     

Mixtures of regression models with incomplete and noisy data

The estimation of the mixtures of regression models is usually based on the normal assumption of components and maximum likelihood estimation of the normal components is sensitive to noise, outliers, or high-leverage points. Missing values are inevitable in many situations and parameter estimates could be biased if the missing values are not handled properly. In this article, we propose the mixtures of regression models for contaminated incomplete heterogeneous data. The proposed models provide robust estimates of regression coefficients varying across latent subgroups even under the presence of missing values. The methodology is illustrated through simulation studies and a real data analysis.     

Variable selection in finite mixture of semi-parametric regression models

Abstract

In this paper we are concerned with variable selection in finite mixture of semiparametric regression models. This task consists of model selection for non parametric component and variable selection for parametric part. Thus, we encountered separate model selections for every non parametric component of each sub model. To overcome this computational burden, we introduced a class of variable selection procedures for finite mixture of semiparametric regression models using penalized approach for variable selection. It is shown that the new method is consistent for variable selection. Simulations show that the performance of proposed method is good, and it consequently improves pervious works in this area and also requires much less computing power than existing methods.     

Adaptive varying-coefficient linear models

Summary. Varying-coefficient linear models arise from multivariate nonparametric regression, non-linear time series modelling and forecasting, functional data analysis, longitudinal data analysis and others. It has been a common practice to assume that the varying coefficients are functions of a given variable, which is often called an index . To enlarge the modelling capacity substantially, this paper explores a class of varying-coefficient linear models in which the index is unknown and is estimated as a linear combination of regressors and/or other variables. We search for the index such that the derived varying-coefficient model provides the least squares approximation to the underlying unknown multidimensional regression function. The search is implemented through a newly proposed hybrid backfitting algorithm. The core of the algorithm is the alternating iteration between estimating the index through a one-step scheme and estimating coefficient functions through one-dimensional local linear smoothing. The locally significant variables are selected in terms of a combined use of the t -statistic and the Akaike information criterion. We further extend the algorithm for models with two indices. Simulation shows that the methodology proposed has appreciable flexibility to model complex multivariate non-linear structure and is practically feasible with average modern computers. The methods are further illustrated through the Canadian mink–muskrat data in 1925–1994 and the pound–dollar exchange rates in 1974–1983.     

Estimation and diagnostics for heteroscedastic nonlinear regression models based on scale mixtures of skew-normal distributions

An extension of some standard likelihood based procedures to heteroscedastic nonlinear regression models under scale mixtures of skew-normal (SMSN) distributions is developed. This novel class of models provides a useful generalization of the heteroscedastic symmetrical nonlinear regression models (Cysneiros et al., 2010), since the random term distributions cover both symmetric as well as asymmetric and heavy-tailed distributions such as skew-t, skew-slash, skew-contaminated normal, among others. A simple EM-type algorithm for iteratively computing maximum likelihood estimates of the parameters is presented and the observed information matrix is derived analytically. In order to examine the performance of the proposed methods, some simulation studies are presented to show the robust aspect of this flexible class against outlying and influential observations and that the maximum likelihood estimates based on the EM-type algorithm do provide good asymptotic properties. Furthermore, local influence measures and the one-step approximations of the estimates in the case-deletion model are obtained. Finally, an illustration of the methodology is given considering a data set previously analyzed under the homoscedastic skew-t nonlinear regression model.     

A three-state disease model with interval-censored data: Estimation and applications to AIDS and cancer

In presence of interval-censored data, we propose a general three-state disease model with covariates. Such data can arise, for example, in epidemiologic studies of infectious disease where both the times of infection and disease onset are not directly observed, or in cancer studies where the time of disease metastasis is known up to a specified interval. The proposed model allows the distributions of the transition times between states to depend on covariates and the time in the previous state. An estimation procedure for the underlying distributions and the model coefficients is suggested with the EM algorithm. The EMS algorithm (Smoothed EM algorithm) is also considered to obtain smooth estimates of the distributions. The proposed method is illustrated with data from an AIDS study and a study of patients with malignant melanoma.     

Estimation and testing for semiparametric mixtures of partially linear models

In this paper, we study the estimation and inference for a class of semiparametric mixtures of partially linear models. We prove that the proposed models are identifiable under mild conditions, and then give a PL–EM algorithm estimation procedure based on profile likelihood. The asymptotic properties for the resulting estimators and the ascent property of the PL–EM algorithm are investigated. Furthermore, we develop a test statistic for testing whether the non parametric component has a linear structure. Monte Carlo simulations and a real data application highlight the interest of the proposed procedures.