Densities for random balanced sampling

A random balanced sample (RBS) is a multivariate distribution with n components X k , each uniformly distributed on [ - 1 , 1 ] , such that the sum of these components is precisely 0. The corresponding vectors X ⇒ 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.