On the problem of the feasibility of optimal stratification points according to Dalenius

Summary

It is well known (see [2]), that the necessary conditions for optimal stratum boundaries by optimal allocation, given by Dalenius and Gurney [1], lead— by sampling fractions q=n/N>0—to nonfeasible solutions, if only one of the conditions n_h≤N_h (h=1 (1) L) is violated.

In [2] it was shown by the example of the logarithmic normal distribution with μ=0, σ=1,5 and L=2 strata, that only for q≤q_c (q_c=critical sampling fraction) the conditions of Dalenius yield feasible solutions. For q>q_C the feasible optimal stratification and allocation results in complete enumeration of the second stratum.

In this work for the logarithmic normal distribution and for L=2 strata there will be shown:

• a) the dependence of the feasible optimal stratification point x₁ on the sampling fraction q=n/N by fixed σ (figure 1). One can see, that this curve has a sharp bend at q=q_c.

• b) the dependence of (q_c, x₁(q_c)) on σ (figure 2). This curve seperates the region of optimal stratification points by Neyman's allocation from the region by complete enumeration.