Hypothesis testing on multivariate data with a special structure is dealt with in this paper. Specifically, the situations considered concern two types of data: directional and compositional. Directional (or spherical) data is comprised of multidimensional random vectors with Euclidean distance from the origin equal to one; that is, the sample space is the unit hypersphere. On the other hand, compositional (or proportional) data contains multidimensional vectors with proportions as coordinates whose sum is one. Both types of data are particularly interesting in various scientific disciplines. For example, directional data is found in astronomy (origin of comets), biology (bird navigation), medicine (variation in the onset of cancer), and meteorology (wind directions). Compositional data can represent percentages of materials in mixtures and is found in disciplines such as geology, chemistry, and archeology, as well as other types of data, such as the results of elections or social activities.

The constrained nature of directional and compositional data raises various interesting problems concerning their analysis and statistical inference through hypothesis testing. The inherent correlation between variables and the lack of traditional Gaussian assumptions make creating new statistical procedures a challenging research task.

Cuesta-Albertos, Cuevas, and Fraiman provide very useful and easy-to-implement techniques for testing the uniformity, sphericity, and homogeneity of two samples of directional and compositional data. The tests proposed are based on one-dimensional projections of the multidimensional random vectors on random directions. The behavior of the tests is studied both theoretically and through simulations. An interesting section applies the proposed tests to an astronomy problem concerning the uniformity of comet orbits.

In conclusion, the ideas discussed in the paper are very interesting, the methods are described in an understandable manner--the advanced mathematical proofs are provided separately in a technical appendix--and the methods are easily programmable in a statistical language such as R. Therefore, I recommend the paper to those researchers who deal with these special types of data.