Biases in galaxy cluster velocity dispersion and mass estimates in the small number of galaxies regime

We present a study of the statistical properties of three velocity dispersion and mass estimators, namely biweight, gapper and standard deviation, in the small number of galaxies regime ($N_{\rm gal} \le 75$). Using a set of 73 numerically simulated galaxy clusters, we characterise the statistical bias and the variance for the three estimators, both in the determination of the velocity dispersion and the dynamical mass of the clusters via the $\sigma-M$ relation. The results are used to define a new set of unbiased estimators, that are able to correct for those statistical biases with a minimal increase of the associated variance. The numerical simulations are also used to characterise the impact of velocity segregation in the selection of cluster members, and the impact of using cluster members within different physical radii from the cluster centre. The standard deviation is found to be the lowest variance estimator. The selection of galaxies within the sub-sample of the most massive galaxies in the cluster introduces a $2\,$\% bias in the velocity dispersion estimate when calculated using a quarter of the most massive cluster members. We also find a dependence of the velocity dispersion estimate on the aperture radius as a fraction of $R_{200}$, consistent with previous results. The proposed set of unbiased estimators effectively provides a correction of the velocity dispersion and mass estimates from all those effects in the small number of cluster members regime. This is tested by applying the new estimators to a subset of simulated observations. Although for a single galaxy cluster the statistical and physical effects discussed here are comparable or slightly smaller than the bias introduced by interlopers, they will be of relevance when dealing with ensemble properties and scaling relations for large cluster samples (Abridged).