Empirical convergence bounds for quasi-monte carlo integration

Abstract

A possibility of a probabilistic approach to a deterministic error estimation procedure of quasi-Monte-Carlo method is presented. Existing estimation methods of the aforementioned estimation are non-constructive. A well-known Koksma-Hlawka inequality includes a constant integrand variation, estimation of which is a more complex problem. Since quasi Monte-Carlo method estimates the integral with the mean integrand value, we can expect that the error distribution (in a probabilistic sense) converges to normal. An additional difficulty, however, is that quasi-random sequences, being treated as randomly generated, are statistically dependent, which complicates the numerical estimation of second moments. Authors propose a new approach to the problem of estimating second order moments, based on new results, derived from a theory of random cubature formulas. Presented numerical examples indicate the superiority of the discussed error estimation method over classical ones. Refs 0. Figs 0. Tables 0.

Keywords:

Monte Carlo and Quasi-Monte Carlo method, confidence interval, random cubature formulas, Haar functions, Sobol sequences