Families of complementary distributions

Each continuous distribution on (0,1), with cumulative distribution function F say, has a complementary distribution which is the distribution with cumulative distribution function F−1. Some basic general properties of complementary distributions are given.A particular focus of this article is then the construction of families of distributions, each based on a given F and indexed by a single additional parameter, which are closed under complementarity; properties of these families of distributions are explored, as are those of a particular special case.