Characterization of the Triply Balanced Matrices with Applications to Survey Sampling

摘要

R x L triply balanced matrices arise in estimating the mean square errors of nonlinear statistics in survey samplings. It is shown that: (1) Any R x L exact triply balanced matrix and an orthogonal array OA(R,L,2,3; lambda) are one and the same object up to a possible notational change of the two symbols of the array; (2) R is a multiple of 8 and L < or = 1/2 R; (3) The problem of the construction of R x L exact triply balanced matrices, 3 < or = L < or = 1/2 R, is completely resolved modulo the existence of Hadamard matrices of order 1/2 R; (4) There is no sequence of R x L matrices which are nearly triply balanced in the sense of Rao and Wu (1985) if R<2L.