Least Absolute Deviation Research Articles

M-estimators of some GARCH-type models; computation and application

In this paper, we consider robust M-estimation of time series models with both symmetric and asymmetric forms of heteroscedasticity related to the GARCH and GJR models. The class of estimators includes least absolute deviation (LAD), Huber's, Cauchy and B-estimator as well as the well-known quasi maximum likelihood estimator (QMLE). Extensive simulations are used to check the relative performance of these estimators in both models and the weighted resampling methods are used to approximate the sampling distribution of M-estimators. Our study indicates that there are estimators that can perform better than QMLE and even outperform robust estimator such as LAD when the error distribution is heavy-tailed. These estimators are also applied to real data sets.

A Monte Carlo Comparison of Regression Estimators When the Error Distribution is Long-Tailed Symmetric

The performances of the ordinary least squares (OLS), modified maximum likelihood (MML), least absolute deviations (LAD), Winsorized least squares (WIN), trimmed least squares (TLS), Theil’s (Theil) and weighted Theil’s (Weighted Theil) estimators are compared under the simple linear regression model in terms of their bias and efficiency when the distribution of error terms is long-tailed symmetric.

Least Absolute Value vs. Least Squares Estimation and Inference Procedures in Regression Models with Asymmetric Error Distributions

A Monte Carlo simulation is used to compare estimation and inference procedures in least absolute value (LAV) and least squares (LS) regression models with asymmetric error distributions. Mean square errors (MSE) of coefficient estimates are used to assess the relative efficiency of the estimators. Hypothesis tests for coefficients are compared on the basis of empirical level of significance and power.

Asymptotics for argmin processes: Convexity arguments

The convexity arguments developed by Pollard [D. Pollard, Asymptotics for least absolute deviation regression estimators, Econometric Theory 7 (1991) 186–199], Hjort and Pollard [N.L. Hjort, D. Pollard, Asymptotics for minimizers of convex processes, 1993 (unpublished manuscript)], and Geyer [C.J. Geyer, On the asymptotics of convex stochastic optimization, 1996 (unpublished manuscript)] are now basic tools for investigating the asymptotic behavior of M -estimators with non-differentiable convex objective functions. This paper extends the scope of convexity arguments to the case where estimators are obtained as stochastic processes. Our convexity arguments provide a simple proof for the asymptotic distribution of regression quantile processes. In addition to quantile regression, we apply our technique to LAD (least absolute deviation) inference for threshold regression.

Seasonal prediction of summertime tropical cyclone activity over the East China Sea using the least absolute deviation regression and the Poisson regression

AbstractIn the present study, we have employed two statistical models to predict summertime (July–September) tropical cyclone (TC) activity over the East China Sea using the least absolute deviation (LAD) regression and the Poisson regression method. Through a lagged correlation analysis of the relationship between the seasonal TC frequency in the target region and several pre‐season environmental parameters for the period 1979–2003, physically interpretable and statistically significant large‐scale environmental parameters were identified as potential predictors. After applying the predictor screening method based on the stepwise regression, three predictors, i.e. sea surface temperature, outgoing long‐wave radiation and 850‐hPa relative vorticity were finally chosen. They are related to the phase transition of El Niño/Southern Oscillation and the strength of the western North Pacific summer monsoon. The correlation coefficient between the predicted and the observed frequency is 0.75 for the LAD model and 0.78 for the Poisson model. The predictions using the two models have a skill improvement of about 60% compared to the reference forecasts. The present study suggests that both models are skillful in predicting summertime TC frequency over the East China Sea with the Poisson model being slightly more skillful than the LAD model. Copyright © 2009 Royal Meteorological Society

Improved residue function and reduced flow dependence in MR perfusion using least-absolute-deviation regularization

Cerebral blood flow (CBF) estimates derived from singular value decomposition (SVD) of time intensity curves from Gadolinium bolus perfusion-weighted imaging are known to underestimate CBF, especially at high flow rates. We report the development of a model-independent delay-invariant deconvolution technique using least-absolute-deviation (LAD) regularization to improve the CBF estimation accuracy. Computer simulations were performed to compare the accuracy of CBF estimates derived from LAD, reformulated SVD (rSVD) and standard SVD (sSVD) techniques. Simulations were performed at image signal-to-noise ratios ranging from 20 to 400, cerebral blood volumes from 1% to 10%, and CBF from 2.5 mL/100 g/min to 176.5 mL/100 g/min to estimate the effect of these parameters on the accuracy of CBF estimation. The LAD method improved the CBF estimation accuracy by up to 32% in gray matter and 23% in white matter compared with rSVD and sSVD methods. LAD method also reduces the systematic bias of rSVD and sSVD methods to baseline SNR while producing more accurate and reproducible residue function calculation than either rSVD or sSVD method. Initial clinical implementation of the method on six representative clinical cases confirm the advantages of the LAD method over rSVD and sSVD methods.

Finite-sample distribution-free inference in linear median regressions under heteroscedasticity and non-linear dependence of unknown form

We construct finite‐sample distribution‐free tests and confidence sets for the parameters of a linear median regression, where no parametric assumption is imposed on the noise distribution. The set‐up studied allows for non‐normality, heteroscedasticity, non‐linear serial dependence of unknown forms as well as for discrete distributions. We consider a mediangale structure—the median‐based analogue of a martingale difference—and show that the signs of mediangale sequences follow a nuisance‐parameter‐free distribution despite the presence of non‐linear dependence and heterogeneity of unknown form. We point out that a simultaneous inference approach in conjunction with sign transformations yield statistics with the required pivotality features—in addition to usual robustness properties. Monte Carlo tests and projection techniques are then exploited to produce finite‐sample tests and confidence sets. Further, under weaker assumptions, which allow for weakly exogenous regressors and a wide class of linear dependence schemes in the errors, we show that the procedures proposed remain asymptotically valid. The regularity assumptions used are notably less restrictive than those required by procedures based on least absolute deviations (LAD). Simulation results illustrate the performance of the procedures. Finally, the proposed methods are applied to tests of the drift in the Standard and Poor's composite price index series (allowing for conditional heteroscedasticity of unknown form).

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.

THE TEMPERATURE DEPENDENT ENTHALPY OF VAPORIZATION OF PURE SUBSTANCES

Recently the universal behavior of the temperature dependent enthalpy of vaporization along with the whole liquid–vapor coexistence curve at equilibrium is described and explained by Roman et al.5 The work (called RWVM relation) succeeds in the combination of the linear relation near the triple point and the renormalization group theory result near the critical point. For the convenience of chemical designs and engineering applications, we report its b values yielding the minimum average absolute deviation (AAD) for 74 pure substances from the NIST web-book and compare the results with other correlations. We find that with an adapted b value, the RWVM relation predicts the data of 47 pure substances with an AAD less than 0.0093, with six more than 0.02 and all less than 0.03 except quantum fluid hydrogen, that is clearly better than other correlations. For most pure substances, b covers the range from 0 to 1. Only one negative value stands for the quantum fluid helium because of its enthalpy of vaporization being experimentally not a monotonic function of the temperature in the range near the triple point.

A Local Linear Least-Absolute-Deviations Estimator of Volatility

An important empirical characteristic of financial time series is that the unconditional distribution of the returns tends to possess heavy tails. This is the motivation for the particular local kernel volatility estimator proposed in this work. Whereas least-square-deviations (LSD) estimators are strongly affected by heavy-tailed distributions, the performance of least-absolute-deviations (LAD) estimators is not. This robustness to heavy tails is evidenced by the more flexible assumptions made on the distributional moments of the observable variable. The simulation examples also highlight the superior performances of the LAD estimator when compared to the LSD estimator under heavy tails conditions. The full nonparametric model is described and the asymptotic properties of the LAD estimator are derived. Extensive Monte Carlo studies strongly suggest that the LAD estimator is asymptotically adaptive to the unknown conditional first moment. The LAD estimator is also used to estimate the volatility of the S&P500 and the BOVESPA returns.

Combining estimating functions for volatility

Accurate estimates of volatility are needed in risk management. Generalized autoregressive conditional heteroscedastic (GARCH) models and random coefficient autoregressive (RCA) models have been used for volatility modelling. Following Heyde [1997. Quasi-likelihood and its Applications. Springer, New York], volatility estimates are obtained by combining two different estimating functions. It turns out that the combined estimating function for the parameter in autoregressive processes with GARCH errors and RCA models contains maximum information. The combination of the least squares (LS) estimating function and the least absolute deviation (LAD) estimating function with application to GARCH model error identification is discussed as an application.

A probe-treatment-reference (PTR) model for the analysis of oligonucleotide expression microarrays

BackgroundMicroarray pre-processing usually consists of normalization and summarization. Normalization aims to remove non-biological variations across different arrays. The normalization algorithms generally require the specification of reference and target arrays. The issue of reference selection has not been fully addressed. Summarization aims to estimate the transcript abundance from normalized intensities. In this paper, we consider normalization and summarization jointly by a new strategy of reference selection.ResultsWe propose a Probe-Treatment-Reference (PTR) model to streamline normalization and summarization by allowing multiple references. We estimate parameters in the model by the Least Absolute Deviations (LAD) approach and implement the computation by median polishing. We show that the LAD estimator is robust in the sense that it has bounded influence in the three-factor PTR model. This model fitting, implicitly, defines an "optimal reference" for each probe-set. We evaluate the effectiveness of the PTR method by two Affymetrix spike-in data sets. Our method reduces the variations of non-differentially expressed genes and thereby increases the detection power of differentially expressed genes.ConclusionOur results indicate that the reference effect is important and should be considered in microarray pre-processing. The proposed PTR method is a general framework to deal with the issue of reference selection and can readily be applied to existing normalization algorithms such as the invariant-set, sub-array and quantile method.

Constrained Least Absolute Deviation Neural Networks

It is well known that least absolute deviation (LAD) criterion or L(1)-norm used for estimation of parameters is characterized by robustness, i.e., the estimated parameters are totally resistant (insensitive) to large changes in the sampled data. This is an extremely useful feature, especially, when the sampled data are known to be contaminated by occasionally occurring outliers or by spiky noise. In our previous works, we have proposed the least absolute deviation neural network (LADNN) to solve unconstrained LAD problems. The theoretical proofs and numerical simulations have shown that the LADNN is Lyapunov-stable and it can globally converge to the exact solution to a given unconstrained LAD problem. We have also demonstrated its excellent application value in time-delay estimation. More generally, a practical LAD application problem may contain some linear constraints, such as a set of equalities and/or inequalities, which is called constrained LAD problem, whereas the unconstrained LAD can be considered as a special form of the constrained LAD. In this paper, we present a new neural network called constrained least absolute deviation neural network (CLADNN) to solve general constrained LAD problems. Theoretical proofs and numerical simulations demonstrate that the proposed CLADNN is Lyapunov stable and globally converges to the exact solution to a given constrained LAD problem, independent of initial values. The numerical simulations have also illustrated that the proposed CLADNN can be used to robustly estimate parameters for nonlinear curve fitting, which is extensively used in signal and image processing.

A Generalized Least Absolute Deviation Method for Parameter Estimation of Autoregressive Signals

This paper proposes a generalized least absolute deviation (GLAD) method for parameter estimation of autoregressive (AR) signals under non-Gaussian noise environments. The proposed GLAD method can improve the accuracy of the estimation of the conventional least absolute deviation (LAD) method by minimizing a new cost function with parameter variables and noise error variables. Compared with second- and high-order statistical methods, the proposed GLAD method can obtain robustly an optimal AR parameter estimation without requiring the measurement noise to be Gaussian. Moreover, the proposed GLAD method can be implemented by a cooperative neural network (NN) which is shown to converge globally to the optimal AR parameter estimation within a finite time. Simulation results show that the proposed GLAD method can obtain more accurate estimates than several well-known estimation methods in the presence of different noise distributions.

基于最小绝对偏差估计的电视预测跟踪

To cope with the occlusion and intersection between targets and the environment, location prediction is employed in the visual tracking system. Target trace is fitted by sliding subsection polynomials based on least absolute deviation (LAD) estimation, and the future location of target is predicted with the fitted trace. Experiment results show that the proposed location prediction algorithm based on LAD estimation has significant robustness advantages over least square (LS) estimation, and it is more effective than LS-based methods in visual tracking.

Robust nonnested hypothesis testing

This article presents robust J and encompassing tests for testing nonnested hypotheses in the presence of outliers in the data. The proposed tests are based on least absolute deviations (LAD) and M-estimators unlike the conventional J and encompassing tests, which are based on least squares or maximum likelihood estimators. These tests can lead to more reliable inference in the presence of outliers than tests based on nonrobust estimators. The tests are illustrated with applications to two economic data sets and an artificially generated data set and compared with their nonrobust counterparts.

Least Absolute Deviation Estimation in Structural Equation Modeling

Least absolute deviation (LAD) is a well-known criterion to fit statistical models, but little is known about LAD estimation in structural equation modeling (SEM). To address this gap, the authors use the LAD criterion in SEM by minimizing the sum of the absolute deviations between the observed and the model-implied covariance matrices. Using Monte Carlo simulations, the authors compare the performance of this LAD estimator along several dimensions (bias, efficiency, convergence, frequencies of improper solutions, and absolute percentage deviation) to the full information maximum likelihood (ML) and unweighted least squares (ULS) estimators in structural equation modeling. The results for LAD are mixed: There are special conditions under which the LAD estimator outperforms ML and ULS, but the simulation evidence does not support a general claim that LAD is superior to ML and ULS in small samples.

Approximate Median Regression via the Box-Cox Transformation

Median regression is used increasingly in many different areas of applications. The usual median regression estimating equations are derived from minimizing the least absolute deviations (LAD). Because they are not a smooth function of the regression parameters, a solution is best obtained using a linear programming algorithm. As an alternative, we propose estimating the median regression parameters via Gaussian estimating equations after applying a Box-Cox transformation to both the outcome and the linear predictor. The proposed estimator is notably more efficient than the standard LAD estimator, albeit with an acknowledged loss of robustness.

Robust Regression Shrinkage and Consistent Variable Selection Through the LAD-Lasso

The least absolute deviation (LAD) regression is a useful method for robust regression, and the least absolute shrinkage and selection operator (lasso) is a popular choice for shrinkage estimation and variable selection. In this article we combine these two classical ideas together to produce LAD-lasso. Compared with the LAD regression, LAD-lasso can do parameter estimation and variable selection simultaneously. Compared with the traditional lasso, LAD-lasso is resistant to heavy-tailed errors or outliers in the response. Furthermore, with easily estimated tuning parameters, the LAD-lasso estimator enjoys the same asymptotic efficiency as the unpenalized LAD estimator obtained under the true model (i.e., the oracle property). Extensive simulation studies demonstrate satisfactory finite-sample performance of LAD-lasso, and a real example is analyzed for illustration purposes.

Robust neural modeling for the cross-sectional analysis of accounting information

The performance of robust artificial neural network models in learning bivariate relationships between accounting magnitudes is assessed in this paper. Predictive performances of a number of modeling paradigms (namely, linear models, log-linear structures, classical ratios and artificial neural networks) are compared with regard to the problem of modeling a number of the most outstanding accounting ratio relations. We conduct a large scale analysis, carried out on a representative Spanish data base. Several model fitting criteria are used for each model class (namely, least squares, weighted least squares, least absolute deviations (LAD), and weighted LAD regressions). Hence, besides the standard (least squares-based) version of each model we test a robust (LAD) counterpart, in principle more adequate for distributions strongly affected by outliers, as typically appear in accounting ratio modeling. Our results strongly suggest that classical ratio models, although much used in practical applications, appear to be largely inadequate for predictive purposes, with linear models (both in their least squares and LAD variants) providing much more adequate specifications. In a number of cases, the linear specification is improved by considering flexible non-linear structures. Neural networks, because of their model-free regression capabilities, let us capture generic non-linearities of unknown form in the modeled relations, as well as providing—when properly trained—robust tools for modeling and prediction of general relationships.