The problem of optimum stratification and allocation withq=n/N>0

Summary

Dalenius/Gurney [1951] published necessary conditions for the stratum boundaries, so that with Neyman's optimal allocation of the sample sizen the variance of the sample mean will become a minimum. They introduced in the variance of the sample mean for the sample sizesn _h the opti mal values according to Neyman and differentiated this variance with respect to the stratum boundaries. Because Neyman's allocation formula yields only feasible solutions forn _h ≤N _h , the conditions ofDalenius result in wrong, i.e. nonfeasible solutions, if one of the restrictionsn _h ≤N _h (h=1 (1) L) is violated.

By the example of a logarithmic normal distribution with μ=0, σ=1,5 forL=2 the behaviour of the Dalenius-Neyman-minimum and that of the feasible minimum will be shown in dependence on the sampling fractionq=n/N and a critical valueq _c will be given. For valuesq>q _c the Dalenius-Neyman-minimum is no longer feasible.

For the same logarithmic normal distribution andL=2 (1) 10 this critical sampling fractionq _c will be given (section 5).

For different values of σ andq the optimal stratum boundaries and sampling fractions are listed in section 6 forL=2;3;4.