Fast integer-valued algorithms for optimal allocations under constraints in stratified sampling

In stratified random sampling, minimizing the variance of a total estimate leads to the optimal allocation. However, in practice, this original method is scarcely appropriate since in many applications additional constraints have to be considered. Three optimization algorithms are presented that solve the integral allocation problem with upper and lower bounds. All three algorithms exploit the fact that the feasible region is a polymatroid and share the important feature of computing the globally optimal integral solution,which generally differs from a solution obtained by rounding. This is in contrast to recent references which, in general, treat the continuous relaxation of the optimization problem. Two algorithms are of polynomial complexity and all of them are fast enough to be applied to complex problems such as the German Census 2011 allocation problem with almost 20,000 strata. © 2015 Elsevier B.V. All rights reserved. 1. Motivation Estimation in Official Statistics, in general, is based on survey sampling methods. Hence, the inference has to be drawn with respect to the sampling design. One very important design is stratified random sampling which is widely used in practice, e.g., in the German Census 2011. The universe is split into strata which generally determine certain subgroups of interest such as regions, house size classes or business codes. In stratified random sampling, the total sample size has then to be allocated to the strata by certain conditions. A finite population U of size N is split into H disjoint strata of size Nh for h = 1, . . . ,H . Next, the total sample of size n has to be split into the stratum specific sample sizes nh which refers to the so-called allocation problem. Within all H strata simple random sampling is applied, either with or without replacement. As an allocation problem, there are several options. In many cases, the total sample size is equally distributed amongst the strata (nh = n/H; equal allocation) or proportionally to the stratum sizes (nh ∝ Nh; proportional allocation). In both cases, the allocations are non-integral and, hence, have to be rounded. In stratified sampling, a well-known unbiased estimator for the total τY of the variable of interest Y in the universe is τ̂ StrRS y = H 