L 1-estimation for varying coefficient models

The varying coefficient model is a useful extension of linear models and has many advantages in practical use. To estimate the unknown functions in the model, the kernel type with local linear least-squares (L ₂) estimation methods has been proposed by several authors. When the data contain outliers or come from population with heavy-tailed distributions, L ₁-estimation should yield better estimators. In this article, we present the local linear L ₁-estimation method and derive the asymptotic distributions of the L ₁-estimators. The simulation results for two examples, with outliers and heavy-tailed distribution, respectively, show that the L ₁-estimators outperform the L ₂-estimators.     

On least absolute values estimation

The resistance of least absolute values (L₁) estimators to outliers and their robustness to heavy-tailed distributions make these estimators useful alternatives to the usual least squares estimators. The recent development of efficient algorithms for L₁ estimation in linear models has permitted their use in practical data analysis. Although in general the L₁ estimators are not unique, there are a number of properties they all share. The set of all L₁ estimators for a given model and data set can be characterized as the convex hull of some extreme estimators. Properties of the extreme estimators and of the L₁-estimate set are considered.     

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.     

Cross-validation Revisited

Data-based choice of the bandwidth is an important problem in kernel density estimation. The pseudo-likelihood and the least-squares cross-validation bandwidth selectors are well known, but widely criticized in the literature. For heavy-tailed distributions, the L₁ distance between the pseudo-likelihood-based estimator and the density does not seem to converge in probability to zero with increasing sample size. Even for normal-tailed densities, the rate of L₁ convergence is disappointingly slow. In this article, we report an interesting finding that with minor modifications both the cross-validation methods can be implemented effectively, even for heavy-tailed densities. For both these estimators, the L₁ distance (from the density) are shown to converge completely to zero irrespective of the tail of the density. The expected L₁ distance also goes to zero. These results hold even in the presence of a strongly mixing-type dependence. Monte Carlo simulations and analysis of the Old Faithful geyser data suggest that if implemented appropriately, contrary to the traditional belief, the cross-validation estimators compare well with the sophisticated plug-in and bootstrap-based estimators.     

Simultaneous sparse model selection and coefficient estimation for heavy-tailed autoregressive processes

We propose a sparse coefficient estimation and automated model selection procedure for autoregressive processes with heavy-tailed innovations based on penalized conditional maximum likelihood. Under mild moment conditions on the innovation processes, the penalized conditional maximum likelihood estimator satisfies a strong consistency, O_P(N^?1/2) consistency, and the oracle properties, where N is the sample size. We have the freedom in choosing penalty functions based on the weak conditions on them. Two penalty functions, least absolute shrinkage and selection operator and smoothly clipped average deviation, are compared. The proposed method provides a distribution-based penalized inference to AR models, which is especially useful when the other estimation methods fail or under perform for AR processes with heavy-tailed innovations [Feigin, Resnick. Pitfalls of fitting autoregressive models for heavy-tailed time series. Extremes. 1999;1:391–422]. A simulation study confirms our theoretical results. At the end, we apply our method to a historical price data of the US Industrial Production Index for consumer goods, and obtain very promising results.     

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.     

Robust multivariate mixture regression models with incomplete data

Multivariate mixture regression models can be used to investigate the relationships between two or more response variables and a set of predictor variables by taking into consideration unobserved population heterogeneity. It is common to take multivariate normal distributions as mixing components, but this mixing model is sensitive to heavy-tailed errors and outliers. Although normal mixture models can approximate any distribution in principle, the number of components needed to account for heavy-tailed distributions can be very large. Mixture regression models based on the multivariate t distributions can be considered as a robust alternative approach. Missing data are inevitable in many situations and parameter estimates could be biased if the missing values are not handled properly. In this paper, we propose a multivariate t mixture regression model with missing information to model heterogeneity in regression function in the presence of outliers and missing values. Along with the robust parameter estimation, our proposed method can be used for (i) visualization of the partial correlation between response variables across latent classes and heterogeneous regressions, and (ii) outlier detection and robust clustering even under the presence of missing values. We also propose a multivariate t mixture regression model using MM-estimation with missing information that is robust to high-leverage outliers. The proposed methodologies are illustrated through simulation studies and real data analysis.     

Change-point detection in a linear model by adaptive fused quantile method

A novel approach to quantile estimation in multivariate linear regression models with change-points is proposed: the change-point detection and the model estimation are both performed automatically, by adopting either the quantile-fused penalty or the adaptive version of the quantile-fused penalty. These two methods combine the idea of the check function used for the quantile estimation and the L₁ penalization principle known from the signal processing and, unlike some standard approaches, the presented methods go beyond typical assumptions usually required for the model errors, such as sub-Gaussian or normal distribution. They can effectively handle heavy-tailed random error distributions, and, in general, they offer a more complex view on the data as one can obtain any conditional quantile of the target distribution, not just the conditional mean. The consistency of detection is proved and proper convergence rates for the parameter estimates are derived. The empirical performance is investigated via an extensive comparative simulation study and practical utilization is demonstrated using a real data example.     

Robust variable selection for the varying coefficient model based on composite L 1–L 2 regression

The varying coefficient model (VCM) is an important generalization of the linear regression model and many existing estimation procedures for VCM were built on L ₂ loss, which is popular for its mathematical beauty but is not robust to non-normal errors and outliers. In this paper, we address the problem of both robustness and efficiency of estimation and variable selection procedure based on the convex combined loss of L ₁ and L ₂ instead of only quadratic loss for VCM. By using local linear modeling method, the asymptotic normality of estimation is driven and a useful selection method is proposed for the weight of composite L ₁ and L ₂. Then the variable selection procedure is given by combining local kernel smoothing with adaptive group LASSO. With appropriate selection of tuning parameters by Bayesian information criterion (BIC) the theoretical properties of the new procedure, including consistency in variable selection and the oracle property in estimation, are established. The finite sample performance of the new method is investigated through simulation studies and the analysis of body fat data. Numerical studies show that the new method is better than or at least as well as the least square-based method in terms of both robustness and efficiency for variable selection.     

Flexible mixture modeling via the multivariate t distribution with?the?Box-Cox transformation: an?alternative to?the?skew-t distribution

Cluster analysis is the automated search for groups of homogeneous observations in a data set. A popular modeling approach for clustering is based on finite normal mixture models, which assume that each cluster is modeled as a multivariate normal distribution. However, the normality assumption that each component is symmetric is often unrealistic. Furthermore, normal mixture models are not robust against outliers; they often require extra components for modeling outliers and/or give a poor representation of the data. To address these issues, we propose a new class of distributions, multivariate t distributions with the Box-Cox transformation, for mixture modeling. This class of distributions generalizes the normal distribution with the more heavy-tailed t distribution, and introduces skewness via the Box-Cox transformation. As a result, this provides a unified framework to simultaneously handle outlier identification and data transformation, two interrelated issues. We describe an Expectation-Maximization algorithm for parameter estimation along with transformation selection. We demonstrate the proposed methodology with three real data sets and simulation studies. Compared with a wealth of approaches including the skew-t mixture model, the proposed t mixture model with the Box-Cox transformation performs favorably in terms of accuracy in the assignment of observations, robustness against model misspecification, and selection of the number of components.     

Some theory and practical uses of trimmed L-moments

Trimmed L-moments, defined by Elamir and Seheult [2003. Trimmed L-moments. Comput. Statist. Data Anal. 43, 299–314], summarize the shape of probability distributions or data samples in a way that remains viable for heavy-tailed distributions, even those for which the mean may not exist. We derive some further theoretical results concerning trimmed L-moments: a relation with the expansion of the quantile function as a weighted sum of Jacobi polynomials; the bounds that must be satisfied by trimmed L-moments; recurrences between trimmed L-moments with different degrees of trimming; and the asymptotic distributions of sample estimators of trimmed L-moments. We also give examples of how trimmed L-moments can be used, analogously to L-moments, in the analysis of heavy-tailed data. Examples include identification of distributions using a trimmed L-moment ratio diagram, shape parameter estimation for the generalized Pareto distribution, and fitting generalized Pareto distributions to a heavy-tailed data sample of computer network traffic.     

The minimum L2 distance estimator for Poisson mixture models

A robust estimator is developed for Poisson mixture models with a known number of components. The proposed estimator minimizes the L₂ distance between a sample of data and the model. When the component distributions are completely known, the estimators for the mixing proportions are in closed form. When the parameters for the component Poisson distributions are unknown, numerical methods are needed to calculate the estimators. Compared to the minimum Hellinger distance estimator, the minimum L₂ estimator can be less robust to extreme outliers, and often more robust to moderate outliers.     

Asymptotics for L1-estimators of regression parameters under heteroscedasticityY

We consider the asymptotic behaviour of L₁ -estimators in a linear regression under a very general form of heteroscedasticity. The limiting distributions of the estimators are derived under standard conditions on the design. We also consider the asymptotic behaviour of the bootstrap in the heteroscedastic model and show that it is consistent to first order only if the limiting distribution is normal.     

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.     

Robust group-Lasso for functional regression model

In this article, we consider the problem of selecting functional variables using the L1 regularization in a functional linear regression model with a scalar response and functional predictors, in the presence of outliers. Since the LASSO is a special case of the penalized least-square regression with L1 penalty function, it suffers from the heavy-tailed errors and/or outliers in data. Recently, Least Absolute Deviation (LAD) and the LASSO methods have been combined (the LAD-LASSO regression method) to carry out robust parameter estimation and variable selection simultaneously for a multiple linear regression model. However, variable selection of the functional predictors based on LASSO fails since multiple parameters exist for a functional predictor. Therefore, group LASSO is used for selecting functional predictors since group LASSO selects grouped variables rather than individual variables. In this study, we propose a robust functional predictor selection method, the LAD-group LASSO, for a functional linear regression model with a scalar response and functional predictors. We illustrate the performance of the LAD-group LASSO on both simulated and real data.     

Rank‐based estimate of four‐parameter logistic model

During drug development, the calculation of inhibitory concentration that results in a response of 50% (IC₅₀) is performed thousands of times every day. The nonlinear model most often used to perform this calculation is a four‐parameter logistic, suitably parameterized to estimate the IC₅₀ directly. When performing these calculations in a high‐throughput mode, each and every curve cannot be studied in detail, and outliers in the responses are a common problem. A robust estimation procedure to perform this calculation is desirable. In this paper, a rank‐based estimate of the four‐parameter logistic model that is analogous to least squares is proposed. The rank‐based estimate is based on the Wilcoxon norm. The robust procedure is illustrated with several examples from the pharmaceutical industry. When no outliers are present in the data, the robust estimate of IC₅₀ is comparable with the least squares estimate, and when outliers are present in the data, the robust estimate is more accurate. A robust goodness‐of‐fit test is also proposed. To investigate the impact of outliers on the traditional and robust estimates, a small simulation study was conducted. Copyright © 2012 John Wiley & Sons, Ltd.     

Weighted L1-estimates for the First-order Bifurcating Autoregressive Model

We developed robust estimators that minimize a weighted L₁ norm for the first-order bifurcating autoregressive model. When all of the weights are fixed, our estimate is an L₁ estimate that is robust against outlying points in the response space and more efficient than the least squares estimate for heavy-tailed error distributions. When the weights are random and depend on the points in the factor space, the weighted L₁ estimate is robust against outlying points in the factor space. Simulated and artificial examples are presented. The behavior of the proposed estimate is modeled through a Monte Carlo study.     

Standard and robust orthogonal regression

A fast routine for converting regression algorithms into corresponding orthogonal regression (OR) algorithms was introduced in Ammann and Van Ness (1988). The present paper discusses the properties of various ordinary and robust OR procedures created using this routine. OR minimizes the sum of the orthogonal distances from the regression plane to the data points. OR has three types of applications. First, L ₂ OR is the maximum likelihood solution of the Gaussian errors-in-variables (EV) regression problem. This L ₂ solution is unstable, thus the robust OR algorithms created from robust regression algorithms should prove very useful. Secondly, OR is intimately related to principal components analysis. Therefore, the routine can also be used to create L ₁, robust, etc. principal components algorithms. Thirdly, OR treats the x and y variables symmetrically which is important in many modeling problems. Using Monte Carlo studies this paper compares the performance of standard regression, robust regression, OR, and robust OR on Gaussian EV data, contaminated Gaussian EV data, heavy-tailed EV data, and contaminated heavy-tailed EV data.     

Lp-estimators in ARCH models

For ergodic ARCH processes, we introduce a one-parameter family of L_p-estimators. The construction is based on the concept of weighted M-estimators. Under weak assumptions on the error distribution, the consistency is established. The asymptotic normality is proved for the special cases p=1 and 2. To prove the asymptotic normality of the L₁-estimator, one needs the existence of a density of the squares of the errors, whereas for the L₂-estimator the existence of fourth moments is assumed. The asymptotic covariance matrix of the estimator depends on the unknown parameter which can be substituted by consistent estimators. For the L₁-estimator we construct a kernel estimator for the unknown density of the square of the errors.     

De l'unicité des estimateurs robustes en régression lorsque le paramètre d'échelle et le paramètre de la régression sont estimés simultanément

Several methods have been suggested to calculate robust M- and G-M -estimators of the regression parameter β and of the error scale parameter σ in a linear model. This paper shows that, for some data sets well known in robust statistics, the nonlinear systems of equations for the simultaneous estimation of β, with an M-estimate with a redescending ψ-function, and σ, with the residual median absolute deviation (MAD), have many solutions. This multiplicity is not caused by the possible lack of uniqueness, for redescending ψ-functions, of the solutions of the system defining β with known σ; rather, the simultaneous estimation of β and σ together creates the problem. A way to avoid these multiple solutions is to proceed in two steps. First take σ as the median absolute deviation of the residuals for a uniquely defined robust M-estimate such as Huber's Proposal 2 or the L₁-estimate. Then solve the nonlinear system for the M-estimate with σ equal to the value obtained at the first step to get the estimate of β. Analytical conditions for the uniqueness of M and G-M-estimates are also given.