We develop a method to extract the shape information of line profiles from discrete kinematic data. The Gauss–Hermite expansion, which is widely used to describe the line-of-sight velocity distributions extracted from absorption spectra of elliptical galaxies, is not readily applicable to samples of discrete stellar velocity measurements, accompanied by individual measurement errors and probabilities of membership. These include data sets on the kinematics of globular clusters and planetary nebulae in the outer parts of elliptical galaxies, as well as giant stars in the Local Group galaxies and the stellar populations of the Milky Way. We introduce two-parameter families of probability distributions describing symmetric and asymmetric distortions of the line profiles from Gaussianity. These are used as the basis of a maximum likelihood estimator to quantify the shape of the line profiles. Tests show that the method outperforms a Gauss–Hermite expansion for discrete data, with a lower limit for the relative gain of ≈2 for sample sizes N ≈ 800. To ensure that our methods can give reliable descriptions of the shape, we develop an efficient test to assess the statistical quality of the obtained fit.
As an application, we turn our attention to the discrete velocity data sets of the dwarf spheroidals (dSphs) of the Milky Way. Sculptor and Fornax have data sets of ≳1000 line-of-sight velocities of probable member stars. In Sculptor, the symmetric deviations are everywhere consistent with velocity distributions more peaked than Gaussian. In Fornax, instead, there is an evolution in the symmetric deviations of the line profile from a peakier to more flat-topped distribution on moving outwards. Although the data sets for Carina and Sextans are smaller, they still comprise several hundreds of stars. Our methods are sensitive enough to detect evidence for velocity distributions more peaked than Gaussian. These results suggest a radially biased orbital structure for the outer parts of Sculptor, Carina and Sextans. On the other hand, tangential anisotropy is favoured in Fornax. This is all consistent with a picture in which Fornax may have had a different evolutionary history to Sculptor, Carina and Sextans.