Modification of the Maximin and <i>ϕ_p</i> (phi) criteria to achieve statistically uniform distribution of sampling points

This paper proposes a sampling technique that delivers robust designs, i.e. point sets selected from a design domain in the shape of a unit hypercube. The designs are guaranteed to provide a <i>statistically uniform</i> point distribution, meaning that every location has the same probability of being selected. Moreover, the designs are <i>sample uniform</i>, meaning that each individual design has its points spread evenly throughout the domain. The <i>sample uniformity</i> (often measured via a <i>discrepancy</i> criterion) is achieved using distance-based criteria (ϕpor Maximin), i.e. criteria normally used in space-filling designs. We show that the standard intersite metrics employed in distance-based criteria (Maximin and ϕp (phi)) do <i>not</i> deliver statistically uniform designs. Similarly, designs optimized via centered <i>L</i>₂ discrepancy or support points are also not statistically uniform. When these designs (after optimization based on intersite distances) are used for Monte Carlo type of integration, their statistical nonuniformity is a serious problem as it may lead to a systematic bias. This paper proposes using a periodic metric to guarantee the statistical uniformity of the family of distance-based designs. The presented designs used as benchmarks in the paper are only taken from the class of Latin Hypercube designs (LHDs), which forces univariate projections to be uniform and improves accuracy in Monte Carlo integration of some functions.