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Determining optimum strata boundaries and sample sizes for skewed population with log-normal distribution. (English) Zbl 1325.62025

Summary: The method of choosing the best boundaries that make strata internally homogenous as far as possible is known as optimum stratification. To achieve this, the strata should be constructed in such a way that the strata variances for the characteristic under study be as small as possible. If the frequency distribution of the study variable \(x\) is known, the optimum strata boundaries (OSB) could be obtained by cutting the range of the distribution at suitable points. If the frequency distribution of \(x\) is unknown, it may be approximated from the past experience or some prior knowledge obtained at a recent study. Many skewed populations have log-normal frequency distribution or may be assumed to follow approximately log-normal frequency distribution. In this article, the problem of finding the OSB and the optimum sample sizes within the stratum for a skewed population with log-normal distribution is studied. The problem of determining the OSB is redefined as the problem of determining optimum strata widths (OSW) and is formulated as a Nonlinear Programming Problem (NLPP) that seeks minimization of the variance of the estimated population mean under Neyman allocation subject to the constraint that the sum of the widths of all the strata is equal to the range of the distribution. The formulated NLPP turns out to be a multistage decision problem that can be solved by dynamic programming technique. A numerical example is presented to illustrate the application and computational details of the proposed method. A comparison study is conducted to investigate the efficiency of the proposed method with other stratification methods, viz., Dalenius and Hodges’ cum \(\sqrt{f}\) method, geometric method by Gunning and Horgan, and Lavallée-Hidiroglou method using Kozak’s algorithm available in the literature. The study reveals that the proposed technique is efficient in minimizing the variance of the estimate of the population mean and is useful to obtain OSB for a skewed population with log-normal frequency distribution.

MSC:

62D05 Sampling theory, sample surveys
90C39 Dynamic programming
90C30 Nonlinear programming
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[1] DOI: 10.1109/34.57681 · Zbl 05112289 · doi:10.1109/34.57681
[2] DOI: 10.1007/BF02960514 · Zbl 0058.12802 · doi:10.1007/BF02960514
[3] DOI: 10.1111/j.1751-5823.2009.00093.x · doi:10.1111/j.1751-5823.2009.00093.x
[4] Baillargeon S., Survey Methodology 37 (1) pp 53– (2011)
[5] Bellman R.E., Dynamic Programming (1957)
[6] DOI: 10.1007/BF01899725 · Zbl 0321.62012 · doi:10.1007/BF01899725
[7] Dalenius T., Skanda Aktuartidskrift 33 pp 203– (1950)
[8] Dalenius T., Skanda Aktuartidskrift 34 pp 133– (1951)
[9] DOI: 10.1080/01621459.1959.10501501 · doi:10.1080/01621459.1959.10501501
[10] Detlefsen R.E., Proceedings of the Survey Research Methods Section pp 214– (1991)
[11] DOI: 10.1214/aoms/1177706096 · Zbl 0122.37101 · doi:10.1214/aoms/1177706096
[12] Gunning P., Survey Methodology 30 (2) pp 159– (2004)
[13] DOI: 10.1214/aoms/1177731356 · Zbl 0060.30104 · doi:10.1214/aoms/1177731356
[14] Hidiroglou M.A., Journal of Business and Economic Statistics 11 pp 397– (1993)
[15] Hillier F.S., Introduction to Operations Research (2010) · Zbl 0155.28202
[16] Khan E.A., Calcutta Statistical Association Bulletin 52 pp 205– (2002) · doi:10.1177/0008068320020510
[17] DOI: 10.1002/(SICI)1520-6750(199702)44:1<69::AID-NAV4>3.0.CO;2-K · Zbl 0882.90129 · doi:10.1002/(SICI)1520-6750(199702)44:1<69::AID-NAV4>3.0.CO;2-K
[18] DOI: 10.1111/1467-842X.00264 · Zbl 1064.62017 · doi:10.1111/1467-842X.00264
[19] Khan M. G.M., Journal of Indian Society of Agricultural Statistics 59 (2) pp 146– (2005)
[20] Khan M. G.M., Survey Methodology 34 (2) pp 205– (2008)
[21] Kozak M., Statistics in Transition 6 (5) pp 797– (2004)
[22] Kozak M., Survey Methodology 32 (2) pp 157– (2006)
[23] Lavallée P., Some Contributions to Optimal Stratification (1987)
[24] Lavallée P., Proceedings of the Section on Survey Research Methods pp 646– (1988)
[25] Lavallée P., Survey Methodology 14 pp 33– (1988)
[26] Lednicki B., Statistics in Transition 6 pp 287– (2003)
[27] Mahalanobis P.C., Sankhya 12 pp 1– (1952)
[28] Nand N., Journal of Applied Statistical Science 16 (4) pp 65– (2008)
[29] DOI: 10.1093/comjnl/7.4.308 · Zbl 0229.65053 · doi:10.1093/comjnl/7.4.308
[30] Niemiro W., Wiadomosci Statystyczne 10 pp 1– (1999)
[31] Rivest L.P., Survey Methodology 28 pp 191– (2002)
[32] DOI: 10.1080/01621459.1968.10480928 · doi:10.1080/01621459.1968.10480928
[33] DOI: 10.1111/j.1467-842X.1963.tb00134.x · Zbl 0113.13705 · doi:10.1111/j.1467-842X.1963.tb00134.x
[34] Singh R., Sankhya 37 pp 109– (1975)
[35] Sweet E.M., Proceedings of the Survey Research Methods Section pp 491– (1995)
[36] Unnithan V. K.G., Sankhya 40 pp 60– (1978)
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