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Robust vertex fitting algorithms are expected to improve the knowledge of the vertex position and of its uncertainty in the presence of mismeasured or misassociated tracks. Such contaminations are likely to happen in real data as well as in realistic detector simulations. This paper describes a simulation study of the sensitivity of two types of robust algorithms: a trimmed least squares estimator and an adaptive estimator. The statistical properties of the algorithms are studied as a function of the source and the level of contamination, and compared to the results obtained with classical least squares estimators. Two typical event topologies are studied: one resembling a high multiplicity primary vertex with a possible contamination from a nearby vertex and one resembling a low multiplicity secondary vertex in a jet.