Measures of location and scale for velocities in clusters of galaxies - A robust approach

Recent observational evidence suggests that few clusters and groups of galaxies have achieved dynamical equilibrium, where a Gaussian distribution of radial velocities might be expected. The canonical estimation techniques, which either assume Gaussian parent populations or clip observed velocity distributions until the Gaussian assumption is satisfied, are not, in general, minimum variance estimators of the kinematic properties of such clusters. In addition, a detailed examination of the local kinematical properties of clusters requires the use of efficient statistical estimators which are insensitive to localized misbehavior in small datasets. For these reasons we suggest that the traditional methods of assigning cluster mean velocities, dispersions, and confidence intervals on these quantities are no longer adequate. In this paper we discuss alternative estimators of the kinematical properties of clusters of galaxies-estimators that are resistant in the presence of outliers, and robust for a broad range of non-Gaussian underlying populations. Because a number of different estimators may be used for any given quantity, we urge a change in the nomenclature to one that does not imply an underlying probabilistic model: we suggest Cᵤ_ for the central location ("mean"), Sᵥ_ for the scale ("dispersion"), and ICᵤ, v_ and ISᵥ_ for the set of confidence intervals about Cᵤ_ and Sᵥ_, respectively. The subscripts u and v indicate the methods used to obtain the sample estimate. Extensive simulations for a number of common situations realizable in small to large samples of cluster radial velocities allow us to identify minimum variance estimators. We also explore the estimation of confidence intervals using the jackknife and bootstrap resampling techniques, and compare these methods to simple formulas based on sample estimates of central location and scale. Our tests reveal that the family of location and scale estimators based on Tukey's biweight prove consistently superior for most applications. Confidence intervals on location based on the biweight also prove superior. Estimators of confidence intervals on scale require resampling- although bootstrapping is preferred, less computationally demanding estimators based on the jackknife of the biweight scale are shown to be adequate for most situations.