STAPRO: 7980 Model 3G pp. 1–6 (col. fig: nil) Please cite this article in press as: Khodsiani, R., Pooladsaz, S., Universal optimal block designs under hub correlation structure. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.06.024. Statistics and Probability Letters xx (xxxx) xxx–xxx Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro Universal optimal block designs under hub correlation structure R. Khodsiani *, S. Pooladsaz Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran article info Article history: Received 25 February 2017 Received in revised form 23 June 2017 Accepted 26 June 2017 Available online xxxx Keywords: Optimal design Binary block design Generalized binary block design Completely symmetric matrix Semibalanced array abstract Universal optimal block designs under general correlation structures are usually difficult to specify theoretically or algorithmically. However, they can sometimes be found for a specific correlation and a particular parameter value. In this paper, a wide class of block designs, binary and non-binary with v treatments and b blocks each of size k is considered and we present a method to construct the universal optimal block designs under the hub correlation when, in this method, b is less than (or equal to) the number of blocks in semibalanced array methods. © 2017 Elsevier B.V. All rights reserved. 1. Introduction 1 The block experiments have been widely used in sciences and engineering, such as agriculture, electric, textile and etc. 2 There is much interest in the optimality of designs of block experiments for different field trials such as interference effects, 3 carryover effects and correlated observations (see Ai et al. (2009), Filipiak and Markiewicz (2014), Zheng (2013)). There are 4 some criteria for assessment of optimality of designs which are used in experimental designs. Kiefer (1975) defined the 5 universal optimality of designs such that the design is optimal under all optimality criteria. Constructing of the universal 6 optimal designs, if exist, is difficult specially when the observations are correlated. 7 The class of block designs is considered as Ω(v, b, k) where v, b and k are the numbers of treatments, blocks and plots 8 per blocks, respectively. Let k = hv + s where h is a nonnegative integer and 0 ≤ s <v. There are different correlation 9 structures for the observations which are considered by many researchers. The first-order autoregressive correlation (AR(1)) 10 is the most usual correlation structure that was considered in many researches. Gill and Shukla (1985) showed that nearest 11 neighbour balanced block designs (NNBDs) are universal optimal in the class of binary and equireplicate block designs under 12 the AR(1) with positive correlations (ρ> 0). Kunert (1987) proved that the designs which are determined by Gill and Shukla 13 (1985), are optimal over all possible block designs when k <v. Das and Dey (1989) introduced the generalized binary block 14 designs (GBDs) for k >v and Pooladsaz and Martin (2005) showed that these designs are universally optimal under the 15 AR(1) with ρ> 0. 16 The circulant correlation is another correlation structure that was introduced by Zhu et al. (2003). They also obtained the 17 D-optimal design for simple linear regression under the circulant correlation with 0 <ρ< 0.5. 18 Under the nearest neighbour correlation structure, the optimal block designs for k ≤ v and v< k ≤ 2v were given by 19 Martin et al. (1993) and Martin (1998), respectively. Rao (1961) introduced the orthogonal array of type II and semibalanced 20 array (SBA). For any k and v, Chai and Majumdar (2000) showed that the universally optimal block designs can be constructed 21 * Corresponding author. E-mail address: razieh.khodsiani@math.iut.ac.ir (R. Khodsiani). http://dx.doi.org/10.1016/j.spl.2017.06.024 0167-7152/© 2017 Elsevier B.V. All rights reserved.