On adaptive linear regression

Ordinary least squares (OLS) is omnipresent in regression modeling. Occasionally, least absolute deviations (LAD) or other methods are used as an alternative when there are outliers. Although some data adaptive estimators have been proposed, they are typically difficult to implement. In this paper, we propose an easy to compute adaptive estimator which is simply a linear combination of OLS and LAD. We demonstrate large sample normality of our estimator and show that its performance is close to best for both light-tailed (e.g. normal and uniform) and heavy-tailed (e.g. double exponential and t ₃) error distributions. We demonstrate this through three simulation studies and illustrate our method on state public expenditures and lutenizing hormone data sets. We conclude that our method is general and easy to use, which gives good efficiency across a wide range of error distributions.     

Outlier Resistant Estimation in Difference-Based Semiparametric Partially Linear Models

Partially linear models are extensions of linear models that include a nonparametric function of some covariate allowing an adequate and more flexible handling of explanatory variables than in linear models. The difference-based estimation in partially linear models is an approach designed to estimate parametric component by using the ordinary least squares estimator after removing the nonparametric component from the model by differencing. However, it is known that least squares estimates do not provide useful information for the majority of data when the error distribution is not normal, particularly when the errors are heavy-tailed and when outliers are present in the dataset. This paper aims to find an outlier-resistant fit that represents the information in the majority of the data by robustly estimating the parametric and the nonparametric components of the partially linear model. Simulations and a real data example are used to illustrate the feasibility of the proposed methodology and to compare it with the classical difference-based estimator when outliers exist.     

Doubly robust weighted composite quantile regression based on SCAD-L2

In this article, a robust variable selection procedure based on the weighted composite quantile regression (WCQR) is proposed. Compared with the composite quantile regression (CQR), WCQR is robust to heavy-tailed errors and outliers in the explanatory variables. For the choice of the weights in the WCQR, we employ a weighting scheme based on the principal component method. To select variables with grouping effect, we consider WCQR with SCAD-L₂ penalization. Furthermore, under some suitable assumptions, the theoretical properties, including the consistency and oracle property of the estimator, are established with a diverging number of parameters. In addition, we study the numerical performance of the proposed method in the case of ultrahigh-dimensional data. Simulation studies and real examples are provided to demonstrate the superiority of our method over the CQR method when there are outliers in the explanatory variables and/or the random error is from a heavy-tailed distribution.     

Shrinkage and penalized estimators in weighted least absolute deviations regression models

In this paper, we consider the estimation problem of the weighted least absolute deviation (WLAD) regression parameter vector when there are some outliers or heavy-tailed errors in the response and the leverage points in the predictors. We propose the pretest and James–Stein shrinkage WLAD estimators when some of the parameters may be subject to certain restrictions. We derive the asymptotic risk of the pretest and shrinkage WLAD estimators and show that if the shrinkage dimension exceeds two, the asymptotic risk of the shrinkage WLAD estimator is strictly less than the unrestricted WLAD estimator. On the other hand, the risk of the pretest WLAD estimator depends on the validity of the restrictions on the parameters. Furthermore, we study the WLAD absolute shrinkage and selection operator (WLAD-LASSO) and compare its relative performance with the pretest and shrinkage WLAD estimators. A simulation study is conducted to evaluate the performance of the proposed estimators relative to that of the unrestricted WLAD estimator. A real-life data example using body fat study is used to illustrate the performance of the suggested estimators.     

Shrinkage estimation for linear regression with ARMA errors

In this paper, we extend the modified lasso of Wang et al. (2007) to the linear regression model with autoregressive moving average (ARMA) errors. Such an extension is far from trivial because new devices need to be called for to establish the asymptotics due to the existence of the moving average component. A shrinkage procedure is proposed to simultaneously estimate the parameters and select the informative variables in the regression, autoregressive, and moving average components. We show that the resulting estimator is consistent in both parameter estimation and variable selection, and enjoys the oracle properties. To overcome the complexity in numerical computation caused by the existence of the moving average component, we propose a procedure based on a least squares approximation to implement estimation. The ordinary least squares formulation with the use of the modified lasso makes the computation very efficient. Simulation studies are conducted to evaluate the finite sample performance of the procedure. An empirical example of ground-level ozone is also provided.     

A new estimator combining the ridge regression and the restricted least squares methods of estimation

It is well-known in the literature on multicollinearity that one of the major consequences of multicollinearity on the ordinary least squares estimator is that the estimator produces large sampling variances, which in turn might inappropriately lead to exclusion of otherwise significant coefficients from the model. To circumvent this problem, two accepted estimation procedures which are often suggested are the restricted least squares method and the ridge regression method. While the former leads to a reduction in the sampling variance of the estimator, the later ensures a smaller mean square error value for the estimator. In this paper we have proposed a new estimator which is based on a criterion that combines the ideas underlying these two estimators. The standard properties of this new estimator have been studied in the paper. It has also been shown that this estimator is superior to both the restricted least squares as well as the ordinary ridge regression estimators by the criterion of mean sauare error of the estimator of the regression coefficients when the restrictions are indeed correct. The conditions for superiority of this estimator over the other two have also been derived for the situation when the restrictions are not correct.     

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.     

A fuzzy robust regression approach applied to bedload transport data

Fuzzy least-square regression can be very sensitive to unusual data (e.g., outliers). In this article, we describe how to fit an alternative robust-regression estimator in fuzzy environment, which attempts to identify and ignore unusual data. The proposed approach concerns classical robust regression and estimation methods that are insensitive to outliers. In this regard, based on the least trimmed square estimation method, an estimation procedure is proposed for determining the coefficients of the fuzzy regression model for crisp input-fuzzy output data. The investigated fuzzy regression model is applied to bedload transport data forecasting suspended load by discharge based on a real world data. The accuracy of the proposed method is compared with the well-known fuzzy least-square regression model. The comparison results reveal that the fuzzy robust regression model performs better than the other models in suspended load estimation for the particular dataset. This comparison is done based on a similarity measure between fuzzy sets. The proposed model is general and can be used for modeling natural phenomena whose available observations are reported as imprecise rather than crisp.     

Robust two parameter ridge M-estimator for linear regression

The problem of multicollinearity and outliers in the data set produce undesirable effects on the ordinary least squares estimator. Therefore, robust two parameter ridge estimation based on M-estimator (ME) is introduced to deal with multicollinearity and outliers in the y-direction. The proposed estimator outperforms ME, two parameter ridge estimator and robust ridge M-estimator according to mean square error criterion. Moreover, a numerical example and a Monte Carlo simulation experiment are presented.     

Relative efficiency of first difference estimator in panel data regression with serially correlated error components

We consider the pooled cross-sectional and time series regression model when the disturbances follow a serially correlated one-way error components. In this context we discovered that the first difference estimator for the regression coefficients is equivalent to the generalized least squares estimator irrespective of the particular form of the regressor matrix when the disturbances are generated by a first order autoregressive process where the autocorrelation is close to unity.     

The influence function of penalized regression estimators

To perform regression analysis in high dimensions, lasso or ridge estimation are a common choice. However, it has been shown that these methods are not robust to outliers. Therefore, alternatives as penalized M-estimation or the sparse least trimmed squares (LTS) estimator have been proposed. The robustness of these regression methods can be measured with the influence function. It quantifies the effect of infinitesimal perturbations in the data. Furthermore, it can be used to compute the asymptotic variance and the mean-squared error (MSE). In this paper we compute the influence function, the asymptotic variance and the MSE for penalized M-estimators and the sparse LTS estimator. The asymptotic biasedness of the estimators make the calculations non-standard. We show that only M-estimators with a loss function with a bounded derivative are robust against regression outliers. In particular, the lasso has an unbounded influence function.     

AN EXAMINATION OF ESTIMATED RESIDUALS IN A REGRESSION WITH AN INFINITE ORDER PARAMETRIC MODEL

We consider a linear regression with the error term that obeys an autoregressive model of infinite order and estimate parameters of the models. The parameters of the autoregressive model should be estimated based on estimated residuals obtained by means of the method of ordinary least squares, because the errors are unobservable. The consistency of the coefficients, variance and spectral density of the model obeyed by the error term is shown. Further, we estimate the coefficients of the linear regression by means of the method of estimated generalized least squares. We also show the consistency of the estimator.

     

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.     

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.     

The Performance of Robust Two-Stage Estimator in Nonlinear Regression with Autocorrelated Error

Some statistics practitioners often ignore the underlying assumptions when analyzing a real data and employ the Nonlinear Least Squares (NLLS) method to estimate the parameters of a nonlinear model. In order to make reliable inferences about the parameters of a model, require that the underlying assumptions, especially the assumption that the errors are independent, are satisfied. However, in a real situation, we may encounter dependent error terms which prone to produce autocorrelated errors. A two-stage estimator (CTS) has been developed to remedy this problem. Nevertheless, it is now evident that the presence of outliers have an unduly effect on the least squares estimates. We expect that the CTS is also easily affected by outliers since it is based on the least squares estimator, which is not robust. In this article, we propose a Robust Two-Stage (RTS) procedure for the estimation of the nonlinear regression parameters in the situation where autocorrelated errors come together with the existence of outliers. The numerical example and simulation study signify that the RTS is more efficient than the NLLS and the CTS methods.     

Robust Sparse Regression with High-Breakdown Value

Penalized least squares estimators are sensitive to the influence of outliers like the ordinary least squares estimator. We propose a sparse regression estimator for robust variable selection and estimation based on a robust initial estimator. It is proven that our estimator has at least the same breakdown value as the initial estimator. Numerical examples are presented to illustrate our method.     

Estimation of regression parameters in the presence of outliers in the response

In this paper, we consider a regression model with non-spherical covariance structure and outliers in the response. The generalized least squares estimator obtained from the full data set is generally not used in the presence of outliers and an estimator based on only the non-outlying observations is preferred. Here we propose as an estimator a convex combination of the full set and the deleted set estimators and compare its performance with the other two.     

A new biased estimator based on ridge estimation

In this paper we introduce a new biased estimator for the vector of parameters in a linear regression model and discuss its properties. We show that our new biased estimator is superior, in the mean square error(mse) sense, to the ordinary least squares (OLS) estimator, the ordinary ridge regression (ORR) estimator and the Liu estimator. We also compare the performance of our new biased estimator with two other special Liu-type estimators proposed in Liu (2003). We illustrate our findings with a numerical example based on the widely analysed dataset on Portland cement.     

Small sample properties of a ridge regression estimator when there exist omitted variables

In this paper, we derive the exact formulae for moments of the ridge regression estimator proposed by Huang (Econ Lett 62:261–264, 1999), when there exist omitted variables. We show the conditions under which the ridge regression estimator has smaller mean squared error (MSE) than the ordinary least squares estimator. Based on the exact formulae for moments, we compare the bias and MSE performances of both estimators by numerical evaluations.     

Robust ridge and robust Liu estimator for regression based on the LTS estimator

In the multiple linear regression analysis, the ridge regression estimator and the Liu estimator are often used to address multicollinearity. Besides multicollinearity, outliers are also a problem in the multiple linear regression analysis. We propose new biased estimators based on the least trimmed squares (LTS) ridge estimator and the LTS Liu estimator in the case of the presence of both outliers and multicollinearity. For this purpose, a simulation study is conducted in order to see the difference between the robust ridge estimator and the robust Liu estimator in terms of their effectiveness; the mean square error. In our simulations, the behavior of the new biased estimators is examined for types of outliers: X-space outlier, Y-space outlier, and X-and Y-space outlier. The results for a number of different illustrative cases are presented. This paper also provides the results for the robust ridge regression and robust Liu estimators based on a real-life data set combining the problem of multicollinearity and outliers.