Robust nonparametric inference in the simple linear regression model

JORGE G. ADROVER AND MATIAS SALIBIÁN-BARRERA

Lisbon, Portugal

Conferencia; International Conference on Robust Statistics; 2006

Instituto Superior Tecnico de Lisboa (The Minerva Research Foundation and European Science Foundation)

The need for robust inference has been noticed and partially addressed in the statistical literature in the past (see for example the book by Barnett and Lewis (1994) and references therein). It is easy to see that confidence intervals (even those based on robust estimators) may become unreliable when the dataare contaminated. More specifically, because of the bias induced on the estimator by the outliers, the confidence interval may no longer be centered at the true value of the parameter. Hence, these confidence intervals might not contain the true parameter and therefore have coverage levels much lower than the nominal ones. Adrover et al. (2004) constructed robust confidence intervals and p-values for the parameters of thelocation and simple linear regression models. These robust confidence intervals are stable, i.e. they maintain coverage levels near the nominal ones even in the presence of outliers and other departures from the parametric model. The approach proposed in Adrover et al. (2004) was to use the maximum bias of theestimator  to optimally widen the confidence intervals in order to guarantee that their coverage level under the presence of outliers is at least the nominal one. Recently, Yohai and Zamar (2004) considered the problem of constructing robust nonparametric confidence intervals for the median when the data distribution is unknown and the data may contain fraction of contamination. By modifying the sign test and its associated confidence interval, they propose confidenceintervals for the median of a continuous distribution that attain nominal coverage level for any other distribution in a contamination neighborhood. In this paper we adapt this nonparametric approach to construct robust confidence intervals for the parameters of the simple linear regression model that do not require the use of bias bounds and that are also able to attain the nominal coverage under the presence of outliers. Referencias J.G. Adrover, J., M. Salibian-Barrera, M. and R.~H. Zamar (2004). Globally robust inference for location and the simple linear regression model. J. of Statist. Planning and Inference, 119, 353--375. B. Barnett and T. Lewis (1994) Outliers in Statistical Data. Wiley, New York. V.J. Yohai and R.H.Zamar (2004). Robust non-parametric inference for the median. Ann. of Statist., Vol. 32, No. 5, 1841--1857. P. Huber (1981).  Robust Statistics. Wiley & Sons, New York.