  Allocation of samples in stratified and/or multistage sampling is one of the
central issues of sampling theory. In a survey of a population often the
constraints for precision of estimators of subpopulations parameters have to be
taken care of during the allocation of the sample. Such issues are often solved
with mathematical programming procedures. In many situations it is desirable to
allocate the sample, in a way which forces the precision of estimates at the
subpopulations level to be both: optimal and identical, while the constraints
of the total (expected) size of the sample (or samples, in two-stage sampling)
are imposed. Here our main concern is related to two-stage sampling schemes. We
show that such problem in a wide class of sampling plans has an elegant
mathematical and computational solution. This is done due to a suitable
definition of the optimization problem, which enables to solve it through a
linear algebra setting involving eigenvalues and eigenvectors of matrices
defined in terms of some population quantities. As a final result we obtain a
very simple and relatively universal method for calculating the subpopulation
optimal and equal-precision allocation which is based on one of the most
standard algorithms of linear algebra (available e.g. in R software).
Theoretical solutions are illustrated through a numerical example based on the
Labour Force Survey. Finally, we would like to stress that the method we
describe, allows to accommodate quite automatically for different levels of
precision priority for subpopulations.
