Given P, a rectangular array with positive elements, a rank one underapproximation for P is given by two positive vectors, say r and s, such that each component of rs' is no greater than the corresponding component of P, whence P can then be written as P=(pi)rs'+(1-pi)D for some constant pi, where the residual matrix D is non-negative. A maximal rank one underapproximation for P is such that pi is maximized over all possible rank one underapproximations for P.
We provide an algorithm for calculating the maximum rank one underapproximation and corresponding pi. We present an explicit expression in the special case of 2 x c tables, and show that the algorithm yields the correct solution in this case.
Note: this report was originally entitled "Technical Report No. B-48, January 1985" in the Columbia Biostatistics Tech Report Series. Due to early font styles, strict and non-strict inequalities may be difficult to distinguish in the scanned version. Magnifying the typeface will help.
Numerical Analysis and Computation
Levin, Bruce, "On Calculating Maximum Rank One Underapproximations for Positive Arrays" (January 1985). Columbia University Biostatistics Technical Report Series. Working Paper 10.