Testing uniformity for the case of a planar unknown support

A new test is proposed for the hypothesis of uniformity on bi‐dimensional supports. The procedure is an adaptation of the “distance to boundary test” (DB test) proposed in Berrendero, Cuevas, & Vázquez‐Grande (2006). This new version of the DB test, called DBU test, allows us (as a novel, interesting feature) to deal with the case where the support S of the underlying distribution is unknown. This means that S is not specified in the null hypothesis so that, in fact, we test the null hypothesis that the underlying distribution is uniform on some support S belonging to a given class ${\cal C}$ . We pay special attention to the case that ${\cal C}$ is either the class of compact convex supports or the (broader) class of compact λ‐convex supports (also called r ‐convex or α‐convex in the literature). The basic idea is to apply the DB test in a sort of plug‐in version, where the support S is approximated by using methods of set estimation. The DBU method is analysed from both the theoretical and practical point of view, via some asymptotic results and a simulation study, respectively. The Canadian Journal of Statistics 40: 378–395; 2012 © 2012 Statistical Society of Canada