Densities for random balanced sampling

A random balanced sample (RBS) is a multivariate distribution with n components Xk, each uniformly distributed on [-1,1], such that the sum of these components is precisely 0. The corresponding vectors lie in an (n-1)-dimensional polytope M(n). We present new methods for the construction of such RBS via densities over M(n) and these apply for arbitrary n. While simple densities had been known previously for small values of n (namely 2,3, and 4), for larger n the known distributions with large support were fractal distributions (with fractal dimension asymptotic to n as n→∞). Applications of RBS distributions include sampling with antithetic coupling to reduce variance, and the isolation of nonlinearities. We also show that the previously known densities (for n⩽4) are in fact the only solutions in a natural and very large class of potential RBS densities. This finding clarifies the need for new methods, such as those presented here.