Influence Functions of the Spearman and Kendall Correlation Measures

Mathematics Subject Classification (2000) 62G35 · 62F99

Estimators of the multiple correlation coefficient: local robustness and confidence intervals.

Many robust regression estimators are defined by minimizing a measure of spread of the residuals. An accompanying R-2-measure, or multiple correlation coefficient, is then easily obtained. In this paper, local robustness properties of these robust R-2-coefficients axe investigated. It is also shown how confidence intervals for the population multiple correlation coefficient can be constructed in the case of multivariate normality.Cautionary note; High breakdown-point; Influence function; Intervals; Model; Multiple correlation coefficient; R-2-measure; Regression analysis; Residuals; Robustness; Squares regression;

The K-Step Spatial Sign Covariance Matrix

The Sign Covariance Matrix is an orthogonal equivariant estimator of mul- tivariate scale. It is often used as an easy-to-compute and highly robust estimator. In this paper we propose a k-step version of the Sign Covariance Matrix, which improves its e±ciency while keeping the maximal breakdown point. If k tends to infinity, Tyler's M-estimator is obtained. It turns out that even for very low values of k, one gets almost the same e±ciency as Tyler's M-estimator.

Generalizing univariate signed rank statistics for testing and estimating a multivariate location parameter.

We generalize signed rank statistics to dimensions higher than one. This results in a class of orthogonally invariant and distribution free tests that can be used for testing spherical symmetry/location parameter. The corresponding estimator is orthogonally equivariant. Both the test and estimator can be chosen with asymptotic efficiency 1. The breakdown point of the estimator depends only on the scores, not on the dimension of the data. For elliptical distributions, we obtain an affine invariant test with the same asymptotic properties, if the signed rank statistic is applied to standardized data. We also present a method for computing the estimator numerically, and consider a real data example and some simulations. Finally, an application to detection of time-varying signals in spherically symmetric noise is given.Affine invariant tests; Asymptotic normality; Breakdown point; distribution free tests;

On the Optimality of Multivariate S-Estimators

In this paper we maximize the efficiency of a multivariate S-estimator under a constraint on the breakdown point. In the linear regression model, it is known that the highest possible efficiency of a maximum breakdown S-estimator is bounded above by 33% for Gaussian errors. We prove the surprising result that in dimensions larger than one, the efficiency of a maxi- mum breakdown S-estimator of location and scatter can get arbitrarily close to 100%, by an appropriate selection of the loss function.Breakdown point;Multivariate Location and Scatter;Robustness;S-estimator

Robust Estimation of Mean and Dispersion Functions in Extended Generalized Additive Models

Generalized Linear Models are a widely used method to obtain parametric es- timates for the mean function. They have been further extended to allow the re- lationship between the mean function and the covariates to be more flexible via Generalized Additive Models. However the fixed variance structure can in many cases be too restrictive. The Extended Quasi-Likelihood (EQL) framework allows for estimation of both the mean and the dispersion/variance as functions of covari- ates. As for other maximum likelihood methods though, EQL estimates are not resistant to outliers: we need methods to obtain robust estimates for both the mean and the dispersion function. In this paper we obtain functional estimates for the mean and the dispersion that are both robust and smooth. The performance of the proposed method is illustrated via a simulation study and some real data examples.dispersion;generalized additive modelling;mean regression function;quasilikelihood;M-estimation;P-splines;robust estimation

The breakdown behavior of the maximum likelihood estimator in the logistic regression model.

In this note we discuss the breakdown behavior of the maximum likelihood (ML) estimator in the logistic regression model. We formally prove that the ML-estimator never explodes to infinity, but rather breaks down to zero when adding severe outliers to a data set. An example confirms this behavior. (C) 2002 Published by Elsevier Science B.V.breakdown point; logistic regression; maximum likelihood; robust estimation; generalized linear-models; robustness; existence; fits;

Robust estimation for ordinal regression.

Ordinal regression is used for modelling an ordinal response variable as a function of some explanatory variables. The classical technique for estimating the unknown parameters of this model is Maximum Likelihood (ML). The lack of robustness of this estimator is formally shown by deriving its breakdown point and its influence function. To robustify the procedure, a weighting step is added to the Maximum Likelihood estimator, yielding an estimator with bounded influence function. We also show that the loss in efficiency due to the weighting step remains limited. A diagnostic plot based on the Weighted Maximum Likelihood estimator allows to detect outliers of different types in a single plot.Breakdown point; Diagnostic plot; Influence function; Ordinal regression; Weighted maximum likelihood; Robust distances;

The breakdown behavior of the maximum likelihood estimator in the logistic regression model.

Abstract: In this note we discuss the breakdown behavior of the Maximum Likelihood (ML) estimator in the logistic regression model. We formally prove that the ML-estimator never explodes to infinity, but rather breaks down to zero when adding severe outliers to a data set. Numerical experiments confirm this behavior. As a more robust alternative, a Weighted Maximum Likelihood (WML) estimator will be considered.Model; Data;

On The Predictive Content Of Production Surveys: A Pan-European Study

For over forty years, Business Tendency Surveys have been collected in multiple member states of the European Union. Previous research has studied the predictive accuracy of the expectation variables included in those surveys through bivariate, within-country, Granger-causality tests, which has resulted in mixed conclusions. We extend previous research in various ways, as we (i) explicitly allow for cross-country influences, and (ii) do so using both bivariate and multivariate Granger-causality tests. Specifically, the multivariate El-Himdi and Roy test is adapted to jointly test the forecasting value of multiple production expectation series, to assess whether part of this joint effect is indeed due to cross-country influences, and to determine which countries' expectation series have most "clout" in predicting the production levels in the other member countries, or have higher "receptivity", in that their production levels are Granger-caused by the other countries' expectations.business surveys;cross-correlations;granger causality;production expectations