Two Dimensional Balanced Sampling Plan Excluding Contiguous Units With Block Size Four

Throughout recent years, statistical problems in environmental and ecological research have received a great amount of attention. One particular problem revolves around the collectionand analysis of data from finite populations. Commonly, in environmental and ecological populations, neighboring units, spatially or sequentially ordered, within a finite population may provide similar information. In an attempt to obtain the most informative picture of a population, researchers may not want to collect information from similar units within the obtainedsample. Moreover, when sampling from such populations, the selection of dispersed units may result in a reduction in the variance of estimators. Balanced sampling plans excludingcontiguous units (BSEC) were first introduced in 1988 by Hedayat, Rao and Stufken. These designs can be used in survey sampling when the contiguous units in some ordering provide similar information, such as estimates of population characteristics.Let X = {0,1,…,v-1},if C(X) = (x₀,x₁,…,x_v-1) is a cyclic ordering of X, then x_i and x_i+1 are said to be contiguous points for 0≤i ,≤v - 2, as are x₀ and x_v-1.A one-dimensional k-sized balanced sampling plan excluding contiguous units is a pair (X,β), where X is a set of v points that has a cyclic ordering C(X), andβis a collection of k-subsets of X called blocks such that any two contiguous points do not appear together in any block while any two noncontiguous points appear together in exactlyλblocks.This design is denoted by 1-BSEC(v, k,λ).Hedayat, Rao and Stufken (1988, [1,2]) established that for k = 3,4, a 1-BSEC(v,k,λ) exists for someλif v≥3k. Stufken and Wright (2001, [22]) established that for k = 5,6,7 and (k, v)≠(7,22), a l-BSEC(v, k,λ) exists for someλif and only if v≥3k + 1. But the solutions give the values ofλthat are very large, while one typically prefers designs with few blocks and hence smaller values ofλ. Colbourn and Ling (1998,1999) established the necessary and sufficient conditions for the existence of 1-BSEC(v, k,λ) for k = 3,4 for fixedλ(see,[19][23]).The idea of two dimensional BSECs was suggested by Hedayat,et al. Here two dimensionalmeans that the set of points, say Z_n×Z_m, is arranged in two dimensions, and the points (x - 1,y), (x + 1,y), (x,y - 1), and (x,y + 1) (reducing the sums mod n and m in the first and second coordinates respectively) are said to be 2-contiguous to the point (x, y). In fact, these points are arrange in a torus.A 2-BSEC(n, m, k,λ) is a pair (X,β), where X = Z_n×Z_m and B is a collection of k-subsets of X called blocks such that any two 2-contiguous points do not appear together in any block while any two points that are not 2-contiguous appear together in exactly A blocks.Bryant, Chang, Rodger and Wei [26] discussed two-dimensional k-sized balanced samplingplan excluding contiguous units, and the existence problem of 2-BSECs was completely solved for the caseλ= 1 and k = 3. However the existence problem for 2-BSEC(m, n, 4,λ) is far from complete.In this paper, we study the constructions of 2-BSECs and the existence of 2-BSECs with block size four. The whole thesis is divided into 4 chapters.Chapter 1 In this chapter, we introduce the survey sampling backgrounds of the balancedsampling plan excluding contiguous units, and present the formal definition of balanced sampling plan excluding contiguous units and give some known results.Chapter 2 We introduce the definitions and notations of GDD, IGDD and HGDD, and give some results of their existence. Moreover, we give the definition, properties and some results of BSEC*, which will play an important role in the later constructions.Chapter 3 Although there are some known constructions of 2-BSECs, these constructionsaren't applicable to other cases of 2-BSECs with block size four. Applying GDD, IGDD, HGDD and BSEC*, we give four general constructions of 2-BSECs. Further, we obtain some constructions of 2-BSECs with block size four.Chapter 4 Applying the constructions in chapter 3 and the directed constructions of some small order, we obtain some results of 2-BSECs with block size four. The main result are:(1) Let m≡8 (mod 24), and n∈{4,7,10,13}, there exists a 2-BSEC(m, n, 4,1). Let m≡8 (mod 24), n≡16 (mod 48), there exists a 2-BSEC(m, n, 4,1).(2) Let n≥4, n≠6, and m≡0 (mod 4), there exists a 2-BSEC(m, n, 4,3).(3) Let m, n≡3 (mod 4) and m, n > 3, there exists a 2-BSEC (m, n, 4,3).(4) (i) Let m≡1 (mod 4), n≡9 (mod 16), and m≠33,49, there exists a 2- BSEC(m, n, 4,3). (ii) Let m, n≡5 (mod 16), there exists a 2-BSEC(m, n, 4,3), except possibly for (m, n) = (21,21). (iii) Let m, n≡13 (mod 16), there exists a 2-BSEC{m, n, 4,3), except possibly for (m,n) = (29,29). (iv) Let m,n≡10 (mod 16), there exists a 2-BSEC(m, n, 4,3), except possibly for (m, n) = (26,26). etc.