Constructions of optimal orthogonal arrays with repeated rows

We construct orthogonal arrays OA$_{\lambda} (k,n)$ (of strength two) having a row that is repeated $m$ times, where $m$ is as large as possible. In particular, we consider OAs where the ratio $m / \lambda$ is as large as possible; these OAs are termed optimal. We provide constructions of optimal OAs for any $k \geq n+1$, albeit with large $\lambda$. We also study basic OAs; these are optimal OAs in which $\gcd(m,\lambda) = 1$. We construct a basic OA with $n=2$ and $k =4t+1$, provided that a Hadamard matrix of order $8t+4$ exists. This completely solves the problem of constructing basic OAs wth $n=2$, modulo the Hadamard matrix conjecture

Bounds for orthogonal arrays with repeated rows

In this expository paper, we mainly study orthogonal arrays (OAs) of strength two having a row that is repeated $m$ times. It turns out that the Plackett-Burman bound (\cite{PB}) can be strengthened by a factor of $m$ for orthogonal arrays of strength two that contain a row that is repeated $m$ times. This is a consequence of a more general result due to Mukerjee, Qian and Wu \cite{Muk} that applies to orthogonal arrays of arbitrary strength $t$. We examine several proofs of the Plackett-Burman bound and discuss which of these proofs can be strengthened to yield the aforementioned bound for OAs of strength two with repeated rows. We also briefly discuss related bounds for $t$-designs, and OAs of strength $t$, when $t > 2$

On the construction of nested space-filling designs

Nested space-filling designs are nested designs with attractive low-dimensional stratification. Such designs are gaining popularity in statistics, applied mathematics and engineering. Their applications include multi-fidelity computer models, stochastic optimization problems, multi-level fitting of nonparametric functions, and linking parameters. We propose methods for constructing several new classes of nested space-filling designs. These methods are based on a new group projection and other algebraic techniques. The constructed designs can accommodate a nested structure with an arbitrary number of layers and are more flexible in run size than the existing families of nested space-filling designs. As a byproduct, the proposed methods can also be used to obtain sliced space-filling designs that are appealing for conducting computer experiments with both qualitative and quantitative factors.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1229 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org