Optimal allocation of the sample size to strata under box constraints. (English) Zbl 1238.62007
Summary: In stratified random sampling without replacement boundary conditions, such as the sample sizes within strata shall not exceed the population sizes in the respective strata, have to be considered. H. Stenger and S. Gabler [Metrika, 61, No. 2, 137–156 (2005; Zbl 1079.62013)] have shown a solution that satisfies upper boundaries of sample fractions within the strata. However, in modern applications one may wish to guarantee also minimal sampling fractions within strata in order to allow for reasonable separate estimations. In this paper, an optimal allocation in the J. Neyman [J. R. Stat. Soc. 97, 558–606 (1934; JFM 61.1310.02)] and A.A. Tschuprov [Metron 2, 461–493, 646–683 (1923)] sense is developed which satisfies upper and lower bounds of the sample sizes within strata. Further, a stable algorithm is given which ensures optimality. The resulting sample allocation enables users to bound design weights within stratified random sampling while considering optimality in allocation.
MSC:
| 62D05 | Sampling theory, sample surveys |
| 65C60 | Computational problems in statistics (MSC2010) |
Software:
RReferences:
| [1] | Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, UK · Zbl 1058.90049 |
| [2] | Neyman J (1934) On the two different aspects of the representative method: the method of stratified sampling and the method of purposive selection. J R Stat Soc 97: 558–606 · JFM 61.1310.02 · doi:10.2307/2342192 |
| [3] | R Development Core Team (2009) R: a language and environment for statistical computing. R foundation for statistical computing, Vienna, ISBN 3-900051-07-0 |
| [4] | Schneeberger H (1993) Unzulässige optimale Schichtung und Aufteilung. All Stat Arch 77: 406–416 |
| [5] | Stefanov SM (2008) Convex separable minimization with box constraints. In: Sixth international congress on industrial applied mathematics (ICIAM07) and GAMM annual meeting. Wiley |
| [6] | Stenger H, Gabler S (2005) Combining random sampling and census strategies, justification of inclusion probabilities equal to 1. Metrika 61: 137–156 · Zbl 1079.62013 · doi:10.1007/s001840400328 |
| [7] | Tschuprov AA (1923) On the mathematical expectation of the moments of frequency distributions in the case of correlated observations. Metron 2(461–493): 646–683 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
