On bayesian robustness: an asymptotic approach

This paper presents a new asymptotic approach to study the robustness of Bayesian inference to changes on the prior distribution. We study the robustness of the posterior density score function when the uncertainty about the prior distribution has been restated as a problem of uncertainty about the model parametrization. Classical robustness tools, such as the influence function and the maximum bias function, are defined for uniparametric models and calculated for the location case. Possible extensions to other models are also briefly discussed

Uniform asymptotics for robust location estimates when the scale is unknown

Most asymptotic results for robust estimates rely on regularity conditions that are difficult to verify in practice. Moreover, these results apply to fixed distribution functions. In the robustness context the distribution of the data remains largely unspecified and hence results that hold uniformly over a set of possible distribution functions are of theoretical and practical interest. Also, it is desirable to be able to determine the size of the set of distribution functions where the uniform properties hold. In this paper we study the problem of obtaining verifiable regularity conditions that suffice to yield uniform consistency and uniform asymptotic normality for location robust estimates when the scale of the errors is unknown. We study M-location estimates calculated with an S-scale and we obtain uniform asymptotic results over contamination neighborhoods. Moreover, we show how to calculate the maximum size of the contamination neighborhoods where these uniform results hold. There is a trade-off between the size of these neighborhoods and the breakdown point of the scale estimate.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Statistics (http://www.imstat.org/aos/) at http://dx.doi.org/10.1214/00905360400000054

Robust nonparametric inference for the median

We consider the problem of constructing robust nonparametric confidence intervals and tests of hypothesis for the median when the data distribution is unknown and the data may contain a small fraction of contamination. We propose a modification of the sign test (and its associated confidence interval) which attains the nominal significance level (probability coverage) for any distribution in the contamination neighborhood of a continuous distribution. We also define some measures of robustness and efficiency under contamination for confidence intervals and tests. These measures are computed for the proposed procedures.Comment: Published at http://dx.doi.org/10.1214/009053604000000634 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Discussion: Conditional growth charts

Discussion of Conditional growth charts [math.ST/0702634]Comment: Published at http://dx.doi.org/10.1214/009053606000000669 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org