Statistica Sinica 13(2003), 1-17

THE DIMENSION DISTRIBUTION AND

QUADRATURE TEST FUNCTIONS

Art B. Owen

Stanford University

Abstract: This paper introduces the dimension distribution for a square integrable function $f$ on $[0,1]^s$. The dimension distribution is used to relate several definitions of the effective dimension of a function. Functions of low effective dimension can be easy to integrate numerically. Many commonly considered quadrature test functions are sums or products of univariate functions, and as a result have particularly simple dimension distributions. Recently some high dimensional isotropic integrals have been successfully treated by quasi-Monte Carlo methods. We show numerically that one such function in $25$ dimensions is very nearly a superposition of functions of $3$ or fewer variables, explaining the success of QMC on that problem. A new result shows that certain isotropic polynomials of degree $n$ generate integrands that are exact superpositions of functions of $n$ or fewer variables.

Key words and phrases: Discrepancy, effective dimension, isotropic integrand, quasi-Monte Carlo.