# An Intermediate Bound on the Star Discrepancy

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 23)

## Abstract

Let $${P}_{n}(z)$$ denote the point set of an n-point rank-1 lattice rule with generating vector $$z$$. A criterion used to assess the ‘goodness’ of the point set $${P}_{n}(z)$$ is the star discrepancy, $${D}^{{_\ast}}({P}_{n}(z))$$. As calculating the star discrepancy is an NP-hard problem, then it is usual to work with bounds on it. In particular, it is known that the following two bounds hold:
$${D}^{{_\ast}}({P}_{ n}(z)) \leq 1 - {(1 - 1/n)}^{d} + T(z,n) \leq 1 - {(1 - 1/n)}^{d} + R(z,n)/2,$$
where d is the dimension and the quantities $$T(z,n)$$ and $$R(z,n)$$ are defined in the paper. Here we provide an intermediate bound on the star discrepancy by introducing a new quantity $$W(z,n)$$ which satisfies
$$T(z,n) \leq W(z,n) \leq R(z,n)/2.$$
Like $$R(z,n)$$, the quantity $$W(z,n)$$ may be calculated to a fixed precision in O(nd) operations. A component-by-component construction based on $$W(z,n)$$ is analysed. We present the results of numerical calculations which indicate that values of $$W(z,n)$$ are much closer to $$T(z,n)$$ than to $$R(z,n)/2$$.

## Notes

### Acknowledgements

This work was carried out when the author was a Visiting Fellow in the School of Mathematics and Statistics at the University of New South Wales. The author acknowledges the hospitality received and gives particular thanks to Dr Frances Kuo and Professor Ian Sloan. The author also thanks Dr Vasile Sinescu for his useful comments on an earlier version of this paper.

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