Advances in Applied Probability

Approximation of bounds on mixed-level orthogonal arrays

Ali Devin Sezer and Ferruh Özbudak

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Mixed-level orthogonal arrays are basic structures in experimental design. We develop three algorithms that compute Rao- and Gilbert-Varshamov-type bounds for mixed-level orthogonal arrays. The computational complexity of the terms involved in the original combinatorial representations of these bounds can grow fast as the parameters of the arrays increase and this justifies the construction of these algorithms. The first is a recursive algorithm that computes the bounds exactly, the second is based on an asymptotic analysis, and the third is a simulation algorithm. They are all based on the representation of the combinatorial expressions that appear in the bounds as expectations involving a symmetric random walk. The Markov property of the underlying random walk gives the recursive formula to compute the expectations. A large deviation (LD) analysis of the expectations provides the asymptotic algorithm. The asymptotically optimal importance sampling (IS) of the same expectation provides the simulation algorithm. Both the LD analysis and the construction of the IS algorithm use a representation of these problems as a sequence of stochastic optimal control problems converging to a limit calculus of a variations problem. The construction of the IS algorithm uses a recently discovered method of using subsolutions to the Hamilton-Jacobi-Bellman equations associated with the limit problem.

Article information

Adv. in Appl. Probab., Volume 43, Number 2 (2011), 399-421.

First available in Project Euclid: 21 June 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05B15: Orthogonal arrays, Latin squares, Room squares 62K99: None of the above, but in this section 65C05: Monte Carlo methods
Secondary: 93E20: Optimal stochastic control 49L99: None of the above, but in this section

Mixed-level orthogonal array Rao bound Gilbert-Varshamov bound error block code counting importance sampling large deviation optimal control asymptotic analysis subsolution approach Hamilton-Jacobi-Bellman equation


Devin Sezer, Ali; Özbudak, Ferruh. Approximation of bounds on mixed-level orthogonal arrays. Adv. in Appl. Probab. 43 (2011), no. 2, 399--421.

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