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Gain in precision by optimum stratification and optimum allocation in dependence on the sampling fraction

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Zusammenfassung

Das Problem der optimalen Schichtung bei Neyman'scher Aufteilung wurde in der Stichprobentheorie bisher fast ausschließlich näherungsweise und für den einfacheren Fall vernachlässigbaren Auswahlsatzes\(q = \tfrac{n}{N} = 0\) untersucht. Der Grund liegt in der Schwierigkeit der Lösung der in (2) angegebenen notwendigen Bedingungen für die optimalen Schichtgrenzen. Durch Zurückführung des Problems der optimalen Schichtung und Neyman'schen Aufteilung auf ein nichtlineares Programm kann in der vorliegenden Arbeit der Genauigkeitsgewinn durch Schichtung und Aufteilung gegenüber einfacher Zufallsauswahl in Abhängigkeit von q untersucht werden. Dabei zeigt sich, daß bei Überschreiten eines kritischen Auswahlsatzes q=qc Totalerhebung in Schichten eintritt; die Lösung nach (2) würde in diesem Fall zu falschen Ergebnissen führen.

Summary

The problem of optimum stratification and Neyman allocation in sampling techniques was investigated until now mainly approximately and for simplification with sampling fraction\(q = \tfrac{n}{N} = 0\). The reason is the difficulty of solving the necessary conditions for the optimum stratum boundaries, given in (2). By reducing the problem of optimum stratification and Neyman allocation to a nonlinear program, in this paper the gain in precision by stratification and allocation against simple random sampling in dependence on q is investigated. It is shown that if q is greater than a critical value qc, there exists total enumeration in strata; the solution according to (2) would then give wrong results.

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Authors
  1. H. Schneeberger
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  2. D. Drefahl
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Schneeberger, H., Drefahl, D. Gain in precision by optimum stratification and optimum allocation in dependence on the sampling fraction. Statistische Hefte 23, 228–237 (1982). https://doi.org/10.1007/BF02933052

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  • DOI: https://doi.org/10.1007/BF02933052

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