Focused Information Criterion and Model Averaging in Quantile Regression

In this article, we study model selection and model averaging in quantile regression. Under general conditions, we develop a focused information criterion and a frequentist model average estimator for the parameters in quantile regression model, and examine their theoretical properties. The new procedures provide a robust alternative to the least squares method or likelihood method, and a major advantage of the proposed procedures is that when the variance of random error is infinite, the proposed procedure works beautifully while the least squares method breaks down. A simulation study and a real data example are presented to show that the proposed method performs well with a finite sample and is easy to use in practice.     

Model selection and model averaging for semiparametric partially linear models with missing data

We study model selection and model averaging in semiparametric partially linear models with missing responses. An imputation method is used to estimate the linear regression coefficients and the nonparametric function. We show that the corresponding estimators of the linear regression coefficients are asymptotically normal. Then a focused information criterion and frequentist model average estimators are proposed and their theoretical properties are established. Simulation studies are performed to demonstrate the superiority of the proposed methods over the existing strategies in terms of mean squared error and coverage probability. Finally, the approach is applied to a real data case.     

Model averaging for multivariate multiple regression models

The theories and applications of model averaging have been developed comprehensively in the past two decades. In this paper, we consider model averaging for multivariate multiple regression models. In order to make use of the correlation information of the dependent variables sufficiently, we propose a model averaging method based on Mahalanobis distance which is related to the correlation of the dependent variables. We prove the asymptotic optimality of the resulting Mahalanobis Mallows model averaging (MMMA) estimators under certain assumptions. In the simulation study, we show that the proposed MMMA estimators compare favourably with model averaging estimators based on AIC and BIC weights and the Mallows model averaging estimators from the single dependent variable regression models. We further apply our method to the real data on urbanization rate and the proportion of non-agricultural population in ethnic minority areas of China.     

Model averaging procedure for varying-coefficient partially linear models with missing responses

This paper is concerned with model averaging procedure for varying-coefficient partially linear models with missing responses. The profile least-squares estimation process and inverse probability weighted method are employed to estimate regression coefficients of the partially restricted models, in which the propensity score is estimated by the covariate balancing propensity score method. The estimators of the linear parameters are shown to be asymptotically normal. Then we develop the focused information criterion, formulate the frequentist model averaging estimators and construct the corresponding confidence intervals. Some simulation studies are conducted to examine the finite sample performance of the proposed methods. We find that the covariate balancing propensity score improves the performance of the inverse probability weighted estimator. We also demonstrate the superiority of the proposed model averaging estimators over those of existing strategies in terms of mean squared error and coverage probability. Finally, our approach is further applied to a real data example.     

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.     

Model averaging for M-estimation

M-estimation is a widely used technique for robust statistical inference. In this paper, we study model selection and model averaging for M-estimation to simultaneously improve the coverage probability of confidence intervals of the parameters of interest and reduce the impact of heavy-tailed errors or outliers in the response. Under general conditions, we develop robust versions of the focused information criterion and a frequentist model average estimator for M-estimation, and we examine their theoretical properties. In addition, we carry out extensive simulation studies as well as two real examples to assess the performance of our new procedure, and find that the proposed method produces satisfactory results.     

Frequentist model averaging for envelope models

The envelope method produces efficient estimation in multivariate linear regression, and is widely applied in biology, psychology, and economics. This paper estimates parameters through a model averaging methodology and promotes the predicting abilities of the envelope models. We propose a frequentist model averaging method by minimizing a cross-validation criterion. When all the candidate models are misspecified, the proposed model averaging estimator is proved to be asymptotically optimal. When correct candidate models exist, the coefficient estimator is proved to be consistent, and the sum of the weights assigned to the correct models, in probability, converges to one. Simulations and an empirical application demonstrate the effectiveness of the proposed method.     

Bayesian model averaging in longitudinal regression models with AR(1) errors with application to a myopia data set

We propose a new iterative algorithm, called model walking algorithm, to the Bayesian model averaging method on the longitudinal regression models with AR(1) random errors within subjects. The Markov chain Monte Carlo method together with the model walking algorithm are employed. The proposed method is successfully applied to predict the progression rates on a myopia intervention trial in children.     

Quantile regression and variable selection for the single-index model

In this paper, we propose a new full iteration estimation method for quantile regression (QR) of the single-index model (SIM). The asymptotic properties of the proposed estimator are derived. Furthermore, we propose a variable selection procedure for the QR of SIM by combining the estimation method with the adaptive LASSO penalized method to get sparse estimation of the index parameter. The oracle properties of the variable selection method are established. Simulations with various non-normal errors are conducted to demonstrate the finite sample performance of the estimation method and the variable selection procedure. Furthermore, we illustrate the proposed method by analyzing a real data set.     

On the least-squares model averaging interval estimator

In many applications of linear regression models, randomness due to model selection is commonly ignored in post-model selection inference. In order to account for the model selection uncertainty, least-squares frequentist model averaging has been proposed recently. We show that the confidence interval from model averaging is asymptotically equivalent to the confidence interval from the full model. The finite-sample confidence intervals based on approximations to the asymptotic distributions are also equivalent if the parameter of interest is a linear function of the regression coefficients. Furthermore, we demonstrate that this equivalence also holds for prediction intervals constructed in the same fashion.     

Estimation and variable selection in single-index composite quantile regression

In this article, a new composite quantile regression estimation approach is proposed for estimating the parametric part of single-index model. We use local linear composite quantile regression (CQR) for estimating the nonparametric part of single-index model (SIM) when the error distribution is symmetrical. The weighted local linear CQR is proposed for estimating the nonparametric part of SIM when the error distribution is asymmetrical. Moreover, a new variable selection procedure is proposed for SIM. Under some regularity conditions, we establish the large sample properties of the proposed estimators. Simulation studies and a real data analysis are presented to illustrate the behavior of the proposed estimators.     

Rank Estimation of Log-Linear Regression with Interval-Censored Data

Interval-censored data arise in a wide variety of research and application fields such as cancer and AIDS studies. In this paper, we study a log-linear regression model when data are subject to interval censoring. We use a U-statistic based on ranks to estimate regression coefficients and establish large sample properties of the estimator. We illustrate the performance of the proposed estimate with simulations and a numerical example.     

Model averaging procedure for partially linear single-index models

This paper is concerned with model selection and model averaging procedures for partially linear single-index models. The profile least squares procedure is employed to estimate regression coefficients for the full model and submodels. We show that the estimators for submodels are asymptotically normal. Based on the asymptotic distribution of the estimators, we derive the focused information criterion (FIC), formulate the frequentist model average (FMA) estimators and construct proper confidence intervals for FMA estimators and FIC estimator, a special case of FMA estimators. Monte Carlo studies are performed to demonstrate the superiority of the proposed method over the full model, and over models chosen by AIC or BIC in terms of coverage probability and mean squared error. Our approach is further applied to real data from a male fertility study to explore potential factors related to sperm concentration and estimate the relationship between sperm concentration and monobutyl phthalate.     

Multivariate regression models for discrete and continuous repeated measurements

A general class of multivariate regression models is considered for repeated measurements with discrete and continuous outcome variables. The proposed model is based on the seemingly unrelated regression model (Zellner, 1962) and an extension of the model of Park and Woolson(1992). The regression parameters of the model are consistently estimated using the two-stage least squares method. When the out come variables are multivariate normal, the two-stage estimator reduces to Zellner’s two-stage estimator. As a special case, we consider the marginal distribution described by Liang and Zeger (1986). Under this this distributional assumption, we show that the two-stage estimator has similar asymptotic properties and comparable small sample properties to Liang and Zeger's estimator. Since the proposed approach is based on the least squares method, however, any distributional assumption is not required for variables outcome variables. As a result, the proposed estimator is more robust to the marginal distribution of outcomes.     

Measuring the Discrepancy of a Parametric Model via Local Polynomial Smoothing

Abstract. In the context of multivariate mean regression, we propose a new method to measure and estimate the inadequacy of a given parametric model. The measure is basically the missed fraction of variation after adjusting the best possible parametric model from a given family. The proposed approach is based on the minimum L ² ‐distance between the true but unknown regression curve and a given model. The estimation method is based on local polynomial averaging of residuals with a polynomial degree that increases with the dimension d of the covariate. For any d ≥ 1 and under some weak assumptions we give a Bahadur‐type representation of the estimator from which ‐consistency and asymptotic normality are derived for strongly mixing variables. We report the outcomes of a simulation study that aims at checking the finite sample properties of these techniques. We present the analysis of a dataset on ultrasonic calibration for illustration.     

Inference about the Regression Parameters Using Median-Ranked Set Sampling

The ranked set samples and median ranked set samples in particular have been used extensively in the literature due to many reasons. In some situations, the experimenter may not be able to quantify or measure the response variable due to the high cost of data collection, however it may be easier to rank the subject of interest. The purpose of this article is to study the asymptotic distribution of the parameter estimators of the simple linear regression model. We show that these estimators using median ranked set sampling scheme converge in distribution to the normal distribution under weak conditions. Moreover, we derive large sample confidence intervals for the regression parameters as well as a large sample prediction interval for new observation. Also, we study the properties of these estimators for small sample setup and conduct a simulation study to investigate the behavior of the distributions of the proposed estimators.     

Assessing large sample bias in misspecified model scenarios with reference to exposure model misspecification in errors-in-variable regression: A new computational approach

In this paper, we develop a numerical method for evaluating the large sample bias in estimated regression coefficients arising due to exposure model misspecification while adjusting for measurement errors in errors-in-variable regression. The application of the proposed method has been demonstrated in the case of a logistic errors-in-variable regression model. The method is based on the combination of Monte-Carlo, numerical and, in some special cases, analytic integration techniques. The proposed method facilitates the investigation of the limiting bias in the estimated regression parameters based on a single data set rather than on repeated data sets as required by the conventional repeated sample method. Simulation studies demonstrate that the proposed method provides very similar estimates of bias in the estimated regression parameters under exposure model misspecification in logistic errors-in-variable regression with a higher degree of precision as compared to the conventional repeated sample method.     

Multiple Regression Model Averaging and the Focused Information Criterion With an Application to Portfolio Choice

ABSTRACT

We consider multiple regression (MR) model averaging using the focused information criterion (FIC). Our approach is motivated by the problem of implementing a mean-variance portfolio choice rule. The usual approach is to estimate parameters ignoring the intention to use them in portfolio choice. We develop an estimation method that focuses on the trading rule of interest. Asymptotic distributions of submodel estimators in the MR case are derived using a localization framework. The localization is of both regression coefficients and error covariances. Distributions of submodel estimators are used for model selection with the FIC. This allows comparison of submodels using the risk of portfolio rule estimators. FIC model averaging estimators are then characterized. This extension further improves risk properties. We show in simulations that applying these methods in the portfolio choice case results in improved estimates compared with several competitors. An application to futures data shows superior performance as well.     

Generalised Rank Regression Estimator with Standard Error Adjusted Lasso

One of the standard variable selection procedures in multiple linear regression is to use a penalisation technique in least‐squares (LS) analysis. In this setting, many different types of penalties have been introduced to achieve variable selection. It is well known that LS analysis is sensitive to outliers, and consequently outliers can present serious problems for the classical variable selection procedures. Since rank‐based procedures have desirable robustness properties compared to LS procedures, we propose a rank‐based adaptive lasso‐type penalised regression estimator and a corresponding variable selection procedure for linear regression models. The proposed estimator and variable selection procedure are robust against outliers in both response and predictor space. Furthermore, since rank regression can yield unstable estimators in the presence of multicollinearity, in order to provide inference that is robust against multicollinearity, we adjust the penalty term in the adaptive lasso function by incorporating the standard errors of the rank estimator. The theoretical properties of the proposed procedures are established and their performances are investigated by means of simulations. Finally, the estimator and variable selection procedure are applied to the Plasma Beta‐Carotene Level data set.     

Regression analysis of current status data with auxiliary covariates and informative observation times

This paper discusses regression analysis of current status failure time data with information observations and continuous auxiliary covariates. Under the additive hazards model, we employ a frailty model to describe the relationship between the failure time of interest and censoring time through some latent variables and propose an estimated partial likelihood estimator of regression parameters that makes use of the available auxiliary information. Asymptotic properties of the resulting estimators are established. To assess the finite sample performance of the proposed method, an extensive simulation study is conducted, and the results indicate that the proposed method works well. An illustrative example is also provided.