Download Abstract inference by Grenander U. PDF

By Grenander U.

ISBN-10: 0471082678

ISBN-13: 9780471082675

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Let A = ((ajJ))denote the transpose of the array OA(N1,nl. m1, g ) and B = ((bi,)),the transpose of the array OA(N2,n2, m2, g ) . Let n = min(n1,n2). Form then x NlN2 array whose elements are ordered pairs. The transpose of the above array can be verified to be an orthogonal array OA(N1N2,n , r n l r n 2 , g). By repeated use of this fact, we get the required result. rn Several results on embedding of orthogonal arrays are available in the literature. By embedding we mean any procedure of adding more columns EXERCISES 47 to an orthogonal array without increasing the number of rows or reducing the strength.

We illustrate the method below. 1. Consider the array OA(l6,5,4. 2: I OOQO OA(16,5,4,2) = 0123 0123 0123 0123 1111 0123 1032 2301 3210 2222 0123 2301 3210 1032 3333 0123 3210 1032 2301 1 . Since m’ = 2 divides rn = 4, we group the four symbols of the first column, say, into two sets, say, {0, 1) and {2,3}. Replacing both the symbols in the first set by 0 and those in the second set by 1, we get an asymmetric orthogonal array OA( 16,5,2 x 44. 2) (in the notation of Chapter 2), which is displayed below: OA(16,5,2 x 44,2)= 0123 0123 0123 0123 m 0123 2301 3210 1032 0123 3210 1032 2301 Orthogonal arrays of strength two obtained through the procedure of collapsing can be found in the catalog of Addelman and Kempthome (1961a) and in Dey (1 985).

1. 1(a) by constructing an OA(27,4,3,3). Take rn = g = 3 in this theorem, and note that the elements of G F(3) are po = 0, p1 = 1, pz = 2. 4. 2. 3, take m = r = 2. 2), can be obtained. 3. 1 that are not covered by any of the theorems of this section. 1. This construction is due to Bose and Bush (1952). 1 yields an OA(64,8,2,4). For some more related results, with a bearing on the construction of twosymbol orthogonal arrays of strength four or more, the reader may see Draper and Mitchell (1968) and the references therein.

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