Ana M. Bianco

Robust testing in the logistic regression model

We are interested in testing hypotheses that concern the parameter of a logistic regression model. A robust Wald-type test based on a weighted Bianco and Yohai [ Bianco, A.M., Yohai, V.J., 1996. Robust estimation in the logistic regression model. In: H. Rieder (Ed) Robust Statistics, Data Analysis, and Computer Intensive Methods In: Lecture Notes in Statistics, vol. 109, Springer Verlag, New York, pp. 17-34] estimator, as implemented by Croux and Haesbroeck [Croux, C., Haesbroeck, G., 2003. Implementing the Bianco and Yohai estimator for logistic regression. Computational Statististics and Data Analysis 44, 273-295], is proposed. The asymptotic distribution of the test statistic is derived. We carry out an empirical study to get a further insight into the stability of the p-value. Finally, a Monte Carlo study is performed to investigate the stability of both the level and the power of the test, for different choices of the weight function.

Robust Estimators under Semi-Parametric Partly Linear Autoregression: Asymptotic Behaviour and Bandwidth Selection

In this article, under a semi-parametric partly linear autoregression model, a family of robust estimators for the autoregression parameter and the autoregression function is studied. The proposed estimators are based on a three-step procedure, in which robust regression estimators and robust smoothing techniques are combined. Asymptotic results on the autoregression estimators are derived. Besides combining robust procedures with M-smoothers, predicted values for the series and detection residuals, which allow to detect anomalous data, are introduced. Robust cross-validation methods to select the smoothing parameter are presented as an alternative to the classical ones, which are sensitive to outlying observations. A Monte Carlo study is conducted to compare the performance of the proposed criteria. Finally, the asymptotic distribution of the autoregression parameter estimator is stated uniformly over the smoothing parameter. Copyright 2007 The Authors Journal compilation 2007 Blackwell Publishing Ltd.

On the asymptotic behavior of one-step estimates in heteroscedastic regression models

The asymptotic distribution of one-step Newton-Raphson estimates is established for a regression model with random carriers and heteroscedastic errors under mild conditions. We also include a class of robust estimates defined as the solution of an implicit equation, such as the MM-estimates.

Resistant estimators in Poisson and Gamma models with missing responses and an application to outlier detection

When dealing with situations in which the responses are discrete or show some type of asymmetry, the linear model is not appropriate to establish the relation between the responses and the covariates. Generalized linear models serve this purpose, since they allow one to model the mean of the responses through a link function, linearly on the covariates. When atypical observations are present in the sample, robust estimators are useful to provide fair estimations as well as to build outlier detection rules. The focus of this paper is to define robust estimators for the regression parameter when missing data possibly occur in the responses. The estimators introduced turn out to be consistent under mild conditions. In particular, resistant methods for Poisson and Gamma models are given. A simulation study allows one to compare the behaviour of the classical and robust estimators, under different contamination schemes. The robustness of the proposed procedures is studied through the influence function, while asymptotic variances are derived from it. Besides, outlier detection rules are defined using the influence function. The procedure is also illustrated by analysing a real data set.

Estimation of the marginal location under a partially linear model with missing responses

In this paper, we consider a semiparametric partially linear regression model where there are missing data in the response. We propose robust Fisher-consistent estimators for the regression parameter, for the regression function and for the marginal location parameter of the response variable. A robust cross-validation method is briefly discussed, although, from our numerical results, the marginal estimators seem not to be sensitive to the bandwidth parameter. Finally, a Monte Carlo study is carried out to compare the performance of the robust proposed estimators among themselves and also with the classical ones, for normal and contaminated samples, under different missing data models. An example based on a real data set is also discussed.

Some results for robust GM-based estimators in heteroscedastic regression models

Abstract In this paper the asymptotic behavior of robust estimates based on GM-estimators when the observations follow a regression model with random carriers and heteroscedastic errors is established. Also one-step Newton–Raphson and reweighted estimates for these models are introduced and their breakdown point is studied. Through a Monte Carlo study their performance for small samples is studied.

Approximate τ—Estimates for Linear Regression Based on Subsampling of Elemental Sets

In this paper we show that approximate τ-estimates for the linear model, computed by the algorithm based on subsampling of elemental subsets, are consistent and with high probability have the same breakdown point that the exactτ-estimate. Then, if these estimates are used as initial values, the reweighted least squares algorithm yields a local minimum of the τ-scale having the same asymptotic distribution and, with high probability, the same breakdown point that the global minimum.

Efficiency of MM- and [tau]-estimates for finite sample size

Suppose that the relative efficiency of a regression estimate with respect to the least squares estimate is measured using a robust scale. Then, it is shown that in the case of normal errors and a finite sample size, it is possible to find MM- and [tau]-estimates which combine high efficiency and high breakdown-point.

On a partly linear autoregressive model with moving average errors

In this paper, we generalise the partly linear autoregression model considered in the literature by including moving average errors when we want to allow a large dependence to the past observations. The strong ergodicity of the process is derived. A consistent procedure to estimate the parametric and nonparametric components is provided together with a test statistic that allows to check the presence of a moving average component in the model. Also, a Monte Carlo study is carried out to check the performance of the given proposals.

Robust estimation in partially linear errors-in-variables models

In many applications of regression analysis, there are covariates that are measured with errors. A robust family of estimators of the parametric and nonparametric components of a structural partially linear errors-in-variables model is introduced. The proposed estimators are based on a three-step procedure where robust orthogonal regression estimators are combined with robust smoothing techniques. Under regularity conditions, it is proved that the resulting estimators are consistent. The robustness of the proposal is studied by means of the empirical influence function when the linear parameter is estimated using the orthogonal M -estimator. A simulation study allows to compare the behaviour of the robust estimators with their classical relatives and a real example data is analysed to illustrate the performance of the proposal.