The Kaplan-Meier (KME) and the parametric maximum likelihood (MLE) estimators of the survival function are compared, when the sample contains several outliers and this was not detected. A measure of relative efficiency of the KME with respect to the MLE is computed by simulation for several sample sizes, percentages of censorship and proportions of outliers in the sample. Results for large sample size are compared with values obtained using approximations to asymptotic variance and bias of the MLE. Exponential and Weibull models are used throughout the paper, both under exponential censoring. The objective is to assess the effect of outliers, under different conditions, on the relative efficiency of the KME respective to the MLE for those popular failure time distributions, in particular when the sample size is finite. It is found that for Weibull samples the effect can be substantial but for exponential samples it is almost negligible.