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A confidence interval for the median of a finite population under unequal probability sampling: a model-assisted approach. (English) Zbl 1120.62007

Summary: This paper presents a method for constructing confidence intervals for the median of a finite population under unequal probability sampling. The model-assisted approach makes use of the \(L_{1}\)-norm to motivate the estimating function which is then used to develop a unified approach to inference which includes not only confidence intervals but hypothesis tests and point estimates. The approach relies on large sample theory to construct the confidence intervals. In cases when second-order inclusion probabilities are not available or easy to compute, the Hartley-Rao variance approximation is employed. Simulations show that the confidence intervals achieve the appropriate confidence level, whether or not the Hartley-Rao variance is employed.

MSC:

62D05 Sampling theory, sample surveys
62F25 Parametric tolerance and confidence regions
62G15 Nonparametric tolerance and confidence regions

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References:

[1] Chambers, R. L.; Dunstan, R., Estimating distribution functions from survey data, Biometrika, 73, 597-604 (1986) · Zbl 0614.62005
[2] Chaudhary, M. A.; Sen, P. K., Reconciliation of asymptotics for unequal probability sampling without replacement, J. Statist. Plann. Inference, 102, 71-81 (2002) · Zbl 1005.62010
[3] Dubnicka, S. R., Confidence intervals for the median of a finite population based on the sign test, Comm. Statist. Theory Methods, 35, 551-569 (2006) · Zbl 1093.62015
[4] Francisco, C. A.; Fuller, W. A., Quantile estimation with a complex survey design, Ann. Statist., 19, 454-469 (1991) · Zbl 0787.62011
[5] Godambe, V. P.; Thompson, M. E., Parameters of superpopulation and survey population: Their relationships and estimation, Internat. Statist. Rev., 54, 127-138 (1986) · Zbl 0612.62011
[6] Hájek, J., Limiting distributions in simple random sampling from a finite population, Publ. Math. Inst. Hungarian Acad. Sci., 5, 361-374 (1960) · Zbl 0102.15001
[7] Hettmansperger, T. P.; McKean, J. W., Robust Nonparametric Statistical Methods (1998), Arnold: Arnold Paris · Zbl 0887.62056
[8] Kuk, A. Y.C., Estimation of distribution functions and medians under sampling with unequal probabilities, Biometrika, 75, 97-103 (1988) · Zbl 0632.62009
[9] Lohr, S. L., Sampling: Design and Analysis (1999), Duxbury Press · Zbl 0967.62005
[10] McCarthy, P. J., Standard error and confidence interval estimation for the median, J. Official Statist., 9, 673-689 (1993)
[11] Meyer, J. S., Outer and inner confidence intervals for finite population quantile intervals, J. Amer. Statist. Assoc., 82, 201-204 (1987) · Zbl 0607.62051
[12] Rao, J. N.K.; Kovar, J. G.; Mantel, H. J., On estimating distribution functions and quantiles from survey data using auxiliary information, Biometrika, 77, 365-375 (1990) · Zbl 0716.62013
[13] Rosén, B., Asymptotic theory for successive sampling with varying probabilities without replacement, I and II, Ann. Math. Statist., 43, 373-397, 748-776 (1972) · Zbl 0246.60018
[14] Särndal, C.-E.; Swensson, B.; Wretman, J., Model Assisted Survey Sampling (1992), Springer: Springer Berlin, pp. 197-205 (Chapter 5)
[15] Sedransk, J.; Meyer, J., Confidence intervals for the quantiles of a finite population: simple random and stratified simple random sampling, J. Roy. Statist. Soc. Ser. B, 40, 239-252 (1978) · Zbl 0386.62009
[16] Woodruff, R. S., Confidence intervals for medians and other position measures, J. Amer. Statist. Assoc., 47, 635-646 (1952) · Zbl 0047.38002
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