A Robust Test for Non-nested Hypotheses

We propose a robust version of Cox-type test statistics for the choice between two non-nested hypotheses. We first show that the influence of small amounts of contamination in the data on the test decision can be very large. Secondly, we build a robust test statistic by using the results on robust parametric tests that are available in the literature and show that the level of the robust test is stable. Finally, we show numerically not only the robustness of this new test statistic but also that its asymptotic distribution is a good approximation of its sample distribution, unlike for the classical test statistic. We apply our results to the choice between a Pareto and an exponential distribution as well as between two competing regressors in the simple linear regression model without intercept.     

Robust weighted one-way ANOVA: Improved approximation and efficiency

A robust test for the one-way ANOVA model under heteroscedasticity is developed in this paper. The data are assumed to be symmetrically distributed, apart from some outliers, although the assumption of normality may be violated. The test statistic to be used is a weighted sum of squares similar to the Welch [1951. On the comparison of several mean values: an alternative approach. Biometrika 38, 330-336.] test statistic, but any of a variety of robust measures of location and scale for the populations of interest may be used instead of the usual mean and standard deviation. Under the commonly occurring condition that the robust measures of location and scale are asymptotically normal, we derive approximations to the distribution of the test statistic under the null hypothesis and to its distribution under alternative hypotheses. An expression for relative efficiency is derived, thus allowing comparison of the efficiency of the test as a function of the choice of the location and scale estimators used in the test statistic. As an illustration of the theory presented here, we apply it to three commonly used robust location–scale estimator pairs: the trimmed mean with the Winsorized standard deviation; the Huber Proposal 2 estimator pair; and the Hampel robust location estimator with the median absolute deviation.     

Adaptive Elastic Net GMM Estimation With Many Invalid Moment Conditions: Simultaneous Model and Moment Selection

This article develops the adaptive elastic net generalized method of moments (GMM) estimator in large-dimensional models with potentially (locally) invalid moment conditions, where both the number of structural parameters and the number of moment conditions may increase with the sample size. The basic idea is to conduct the standard GMM estimation combined with two penalty terms: the adaptively weighted lasso shrinkage and the quadratic regularization. It is a one-step procedure of valid moment condition selection, nonzero structural parameter selection (i.e., model selection), and consistent estimation of the nonzero parameters. The procedure achieves the standard GMM efficiency bound as if we know the valid moment conditions ex ante, for which the quadratic regularization is important. We also study the tuning parameter choice, with which we show that selection consistency still holds without assuming Gaussianity. We apply the new estimation procedure to dynamic panel data models, where both the time and cross-section dimensions are large. The new estimator is robust to possible serial correlations in the regression error terms.     

GMM estimation of spatial autoregressive models in a system of simultaneous equations with heteroskedasticity

This paper proposes a GMM estimation framework for the SAR model in a system of simultaneous equations with heteroskedastic disturbances. Besides linear moment conditions, the proposed GMM estimator also utilizes quadratic moment conditions based on the covariance structure of model disturbances within and across equations. Compared with the QML approach, the GMM estimator is easier to implement and robust under heteroskedasticity of unknown form. We derive the heteroskedasticity-robust standard error for the GMM estimator. Monte Carlo experiments show that the proposed GMM estimator performs well in finite samples.     

The indirect method for stochastic logistic growth models

We developed the indirect method for stochastic logistic growth models involving both birth and death rates in the drift and diffusion coefficients, and not only propose two indirect estimators, but also construct a likelihood ratio-type indirect statistic for testing hypotheses concerning parameters. Simulations show that the proposed two indirect estimators can correct the discretization bias, and the proposed indirect test possesses very good estimated power and size.     

Robust tests for the common principal components model

When dealing with several populations, the common principal components (CPC) model assumes equal principal axes but different variances along them. In this paper, a robust log-likelihood ratio statistic allowing to test the null hypothesis of a CPC model versus no restrictions on the scatter matrices is introduced. The proposal plugs into the classical log-likelihood ratio statistic robust scatter estimators. Using the same idea, a robust log-likelihood ratio and a robust Wald-type statistic for testing proportionality against a CPC model are considered. Their asymptotic distributions under the null hypothesis and their partial influence functions are derived. A small simulation study allows to compare the behavior of the classical and robust tests, under normal and contaminated data.     

A Wald-type test statistic based on robust modified median estimator in logistic regression models

In this paper a new robust estimator, modified median estimator, is introduced and studied for the logistic regression model. This estimator is based on the median estimator considered in Hobza et al. [Robust median estimator in logistic regression. J Stat Plan Inference. 2008;138:3822–3840]. Its asymptotic distribution is obtained. Using the modified median estimator, we also consider a Wald-type test statistic for testing linear hypotheses in the logistic regression model and we obtain its asymptotic distribution under the assumption of random regressors. An extensive simulation study is presented in order to analyse the efficiency as well as the robustness of the modified median estimator and Wald-type test based on it.     

Calibration as estimation

Calibration in macroeconomics involves choosing fre parameters by matching certain moments of simulted models with those of data. We formally examine this method by treating the process of calibration as an econometric estimator. A numerical version of the Mehra-Prescott (1985) economy is the setting for an evaluation of calibration estimators via Monte Carlo methods. While these estimators sometimes have reasonable finite-sample properties they are not robust to mistakes in setting non-free parameters. In contrast, generalized method-of-moments (GMM) estimators have satisfactory finite-sample characteristics, quick convergence, and informational requirements less stringent than those of calibration estimators. In dynamic equilibrium models in which GMM is infeasible we offer some suggestions for improving estimates based on calibration methodology.     

Performance of Robust RA Estimator for Bidimensional Autoregressive Models

The additive AR-2D model has been successfully related to the modeling of satelital images both optic and of radar of synthetic opening. Having in mind the errors that are produced in the process of captation and quantification of the image, an interesting subject, is the robust estimation of the parameters in this model. Besides the robust methods in image models are also applied in some important image processing situations such as segmentation by texture and image restoration in the presence of outliers. This paper is concerned with the development and performance of the robust RA estimator proposed by Ojeda (1998) for the estimation of parameters in contaminated AR-2D models. Here, we implement this estimator and we show by simulation study that it has a better performance than the classic least square estimator and the robust M and GM estimators in an additive outlier contaminated image model.     

Multidimensional Poverty: Measurement,Estimation, and Inference

Calibration in macroeconomics involves choosing fre parameters by matching certain moments of simulted models with those of data. We formally examine this method by treating the process of calibration as an econometric estimator. A numerical version of the Mehra-Prescott (1985) economy is the setting for an evaluation of calibration estimators via Monte Carlo methods. While these estimators sometimes have reasonable finite-sample properties they are not robust to mistakes in setting non-free parameters. In contrast, generalized method-of-moments (GMM) estimators have satisfactory finite-sample characteristics, quick convergence, and informational requirements less stringent than those of calibration estimators. In dynamic equilibrium models in which GMM is infeasible we offer some suggestions for improving estimates based on calibration methodology.     

Indirect inference for fractional time series models

In this paper we present an indirect estimation procedure for (ARFIMA) fractional time series models.The estimation method is based on an ‘incorrect’criterion which does not directly provide a consistent estimator of the parameters of interest,but leads to correct inference by using simulations.

The main steps are the following. First,we consider an auxiliary model which can be easily estimated.Specifically,we choose the finite lag Autoregressive model.Then, this is estimated on the observations and simulated values drawn from the ARFIMA model associated with a given value of the parameters of interest.Finally,the latter is calibrated in order to obtain close values of the two estimators of the auxiliary parameters.

In this article,we describe the estimation procedure and compare the performance of the indirect estimator with some alternative estimators based on the likelihood function by a Monte Carlo study.     

A Robust Score Test for Testing Several Coefficients of Variation with Unknown Underlying Distributions

A parametric robust test is proposed for comparing several coefficients of variation. This test is derived by properly correcting the normal likelihood function according to the technique suggested by Royall and Tsou. The proposed test statistic is asymptotically valid for general random variables, as long as their underlying distributions have finite fourth moments.

Simulation studies and real data analyses are provided to demonstrate the effectiveness of the novel robust procedure.     

Invariant tests based on M-estimators,estimating functions,and the generalized method of moments

We study the invariance properties of various test criteria which have been proposed for hypothesis testing in the context of incompletely specified models, such as models which are formulated in terms of estimating functions (Godambe, 1960) or moment conditions and are estimated by generalized method of moments (GMM) procedures (Hansen, 1982), and models estimated by pseudo-likelihood (Gouriéroux, Monfort, and Trognon, 1984b,c) and M-estimation methods. The invariance properties considered include invariance to (possibly nonlinear) hypothesis reformulations and reparameterizations. The test statistics examined include Wald-type, LR-type, LM-type, score-type, and C(α)?type criteria. Extending the approach used in Dagenais and Dufour (1991), we show first that all these test statistics except the Wald-type ones are invariant to equivalent hypothesis reformulations (under usual regularity conditions), but all five of them are not generally invariant to model reparameterizations, including measurement unit changes in nonlinear models. In other words, testing two equivalent hypotheses in the context of equivalent models may lead to completely different inferences. For example, this may occur after an apparently innocuous rescaling of some model variables. Then, in view of avoiding such undesirable properties, we study restrictions that can be imposed on the objective functions used for pseudo-likelihood (or M-estimation) as well as the structure of the test criteria used with estimating functions and generalized method of moments (GMM) procedures to obtain invariant tests. In particular, we show that using linear exponential pseudo-likelihood functions allows one to obtain invariant score-type and C(α)?type test criteria, while in the context of estimating function (or GMM) procedures it is possible to modify a LR-type statistic proposed by Newey and West (1987) to obtain a test statistic that is invariant to general reparameterizations. The invariance associated with linear exponential pseudo-likelihood functions is interpreted as a strong argument for using such pseudo-likelihood functions in empirical work.     

M-estimators for isotonic regression

In this paper we propose a family of robust estimates for isotonic regression: isotonic M-estimators. We show that their asymptotic distribution is, up to an scalar factor, the same as that of Brunk's classical isotonic estimator. We also derive the influence function and the breakdown point of these estimates. Finally we perform a Monte Carlo study that shows that the proposed family includes estimators that are simultaneously highly efficient under Gaussian errors and highly robust when the error distribution has heavy tails.     

Formulating robust regression estimation as an optimum allocation problem

The Mallows-type estimator, one of the most reasonable bounded influence estimators, often downweights leverage points regardless of the magnitude of the corresponding residual, and this could imply a loss of efficiency. In this article, we consider whether the efficiency of this bounded influence estimator could be improved by regarding both the robust x -distance and the residual size. We develop a new robust procedure based on the ideas of the Mallows-type estimator and the general robust recipe, where data been cleaned by pulling outliers towards their fitted values. Our basic idea is to formulate the robust estimation as an allocation problem, where the objective function is a Huber-type "loss" function, but the pulling resource is restricted. Using a mathematical programming technique, the pulling resource is optimally allocated to influential points <$>({x}_i, y_i)<$> with respect to residual size and given weights, <$>w({x}_i)<$>. Three previously published approaches are compared to our proposal via simulated experiments. In the case of contaminated data by regression outliers and "good" leverage points, the proposed robust estimator is a reasonable bounded influence estimator concerning both efficiency and norm of bias. In addition, the proposed approach offers the potential to establish constraints for the regression parameters and also may potentially provide insight regarding outlier detection.     

Improving the bandwidth-free inference methods by prewhitening

In this paper we consider inference of parameters in time series regression models. In the traditional inference approach, the heteroskedasticity and autocorrelation consistent (HAC) estimation is often involved to consistently estimate the asymptotic covariance matrix of regression parameter estimator. Since the bandwidth parameter in the HAC estimation is difficult to choose in practice, there has been a recent surge of interest in developing bandwidth-free inference methods. However, existing simulation studies show that these new methods suffer from severe size distortion in the presence of strong temporal dependence for a medium sample size. To remedy the problem, we propose to apply the prewhitening to the inconsistent long-run variance estimator in these methods to reduce the size distortion. The asymptotic distribution of the prewhitened Wald statistic is obtained and the general effectiveness of prewhitening is shown through simulations.     

A minimum variance adaptive technique for parameter estimation and hypothesis testing

This report presents numerical results of an approach for parameter estimation and hypothesis testing that does not rely on specific assumptions about the underlying distribution of errors in the measured data. This approach combines robust estimation procedures, the bootstrap method for estimation of parameter uncertainties, permutation techniques for hypothesis testing, and adaptive approaches to estimation in order to obtain the minimum variance estimator or test statistic (within a predefined class) for the data under consideration. The technique produces efficient estimators of central tendency and powerful test statistics, even for small sample sizes. (Portions of this work have been presented in preliminary form (Turkheimer et al., 1996)).     

Inference in a two-parameter generalized order statistics model

As a submodel of generalized order statistics with two unknown model parameters, m-generalized order statistics may serve as a simple model for ordered quantities in a given application. It is shown that the joint distribution of m-generalized order statistics has a representation as a regular exponential family in the model parameters, as it is the case for the comprising model. Utilizing this finding, a minimal sufficient and complete statistic is obtained along with distributional properties. Joint maximum likelihood estimation of the parameters is considered, and strong consistency and asymptotic efficiency of the estimator are established. A test is provided to decide whether a restriction to the submodel is reasonable.     

Quasi maximum likelihood estimation of dynamic panel data models

This article establishes the almost sure convergence and asymptotic normality of levels and differenced quasi maximum likelihood (QML) estimators of dynamic panel data models. The QML estimators are robust with respect to initial conditions, conditional and time-series heteroskedasticity, and misspecification of the log-likelihood. The article also provides an ECME algorithm for calculating levels QML estimates. Finally, it compares the finite-sample performance of levels and differenced QML estimators, the differenced generalized method of moments (GMM) estimator, and the system GMM estimator. The QML estimators usually have smaller— typically substantially smaller—bias and root mean squared errors than the panel data GMM estimators.     

A modified ridge m-estimator for linear regression model with multicollinearity and outliers

The ordinary least-square estimators for linear regression analysis with multicollinearity and outliers lead to unfavorable results. In this article, we propose a new robust modified ridge M-estimator (MRME) based on M-estimator (ME) to deal with the combined problem resulting from multicollinearity and outliers in the y-direction. MRME outperforms modified ridge estimator, robust ridge estimator and ME, according to mean squares error criterion. Furthermore, a numerical example and a Monte Carlo simulation experiment are given to illustrate some of the theoretical results.