Summary
It is known (cf. [1]) that the necessary conditions published by Dalenius and Gurney [2] for best stratum boundaries using optimal allocation yield nonfeasible solutions for values of sampling fractions q=n/N>qC (=critical sampling fraction). For the logarithmic normal distribution with L=2 strata, the dependence of the optimal stratum boundary x1 as a function of q=n/N is published in [3] for different values of σ and q (i. e. q≤qC and q≥qC).
In the following paper the numerical values of the critical sampling fractions qC=qC(L) are presented as a function of the number of strata L for the standard normal distribution and for the logarithmic normal distributions with μ=0 and σ=0,5 (0,5) 2. For q>qC (L) the Dalenius-Neyman solution isn't feasible.
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References
Schneeberger, H.: The problem of optimum stratification and allocation with q=n/N>0; Metrika, probably in vol. 28, part 2.
Dalenius and M. Gurney: The Problem of optimum stratification II; Skand. Aktuarietidskrift (1951), 133–148.
Schneeberger, H. and W. Goller: On the problem of the feasibility of optimal stratification points according to Dalenius. Statistische Hefte, 19 (1979), H. 1, 250–256.
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Schneeberger, H., Drefahl, D. Limits of feasible sampling fractions in optimal stratification. Statistische Hefte 21, 61–65 (1980). https://doi.org/10.1007/BF02932812
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DOI: https://doi.org/10.1007/BF02932812