Homogeneity of marginal distributions for a large number of populations

Assume that a random vector is observed in populations and independent samples of that random vector are available at each population. Assume that and have the same dimension. Our purpose is to test the equality of the marginal distributions of and in the populations when is large compared to the sample sizes. With this aim, we propose and study a test statistic that compares the empirical characteristic functions of the marginal distributions. Under the null, the test statistic is asymptotically free-distributed. An expression of the asymptotic power is also derived, which allows to study the consistency of the test. No assumption is made on the distribution of and , which can be continuous, discrete or mixed; moreover, no assumption is made about moments. A simulation study investigates the finite sample performance of the new test. The proposal is applied to study air pollution levels that are directly related to environmental health, in all countries where observations are available.