Equivalence of Neyman Optimum Allocation for Sampling and Equal Proportions for Apportioning the House of Representative (Updated 11/1/13)
Tommy Wright (U.S. Bureau of the Census) presents a surprising though obvious result that seems to have been unnoticed until now. We demonstrate the equivalence of two well-known problems -- the optimal allocation of the fixed overall sample size n among L strata under stratified random sampling and the optimal allocation of the 435 seats among the 50 states for apportionment of the U.S. House of Representatives following each decennial census. In spite of the strong similarity manifest in the statements of the two problems, they have not been linked and they have well-known but different solutions; one solution is not explicitly exact (Neyman allocation), and the other (equal proportions) is exact. We give explicit exact solutions for both and note that the solutions are equivalent. In fact, we conclude by showing that both problems are special cases of a general problem. The result is significant for stratified random sampling in that it explicitly shows how to minimize sampling error when estimating a total while keeping the final overall sample size fixed at n; this is usually not the case in practice with Neyman allocation where the resulting final overall sample size might be near n + L after rounding. An example reveals that controlled rounding with Neyman allocation does not always lead to the optimum allocation that minimizes variance.