Constraining Arrays of Numbers in Demography

Charles D. Coleman, U.S. Census Bureau

Demographic work often requires constraining arrays of numbers to controls in one or two dimensions. If the results are allowed to take on any nonnegative values, raking solves the problem in one dimension and two-way iterative raking solves it in two dimensions. The problem is more complicated in one dimension if the data can be of any sign, the so-called “plus-minus” problem, which is addressed by generalized raking, which preserves the structure of the data at the cost of a nonunique solution. Since demography is concerned with people, data often have to be rounded to integers. The Cox-Ernst algorithm accomplishes an optimal controlled rounding in two dimensions. The Greatest Mantissa algorithm is a simplified version of it in one dimension. This paper combines into one place all of the above-mentioned problems and techniques. It is written to provide practical guidance to demographers and other practitioners who work with these problems.