Abstract
Thompson (1990) introduced the adaptive cluster sampling design and developed two unbiased estimators, the modified Horvitz-Thompson (HT) and Hansen-Hurwitz (HH) estimators, for this sampling design and noticed that these estimators are not a function of the minimal sufficient statistics. He applied the Rao-Blackwell theorem to improve them. Despite having smaller variances, these latter estimators have not received attention because a suitable method or algorithm for computing them was not available. In this paper we obtain closed forms of the Rao-Blackwell versions which can easily be computed. We also show that the variance reduction for the HH estimator is greater than that for the HT estimator using Rao-Blackwell versions. When the condition for extra samples is \(y > 0\), one can expect some Rao-Blackwell improvement in the HH estimator but not in the HT estimator. Two examples are given.
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References
Smith, D.R., Conroy, M.J., and Brakhage, D.H. (1995) Efficiency of adaptive cluster sampling for estimating density of wintering waterfowl. Biometrics, 51, 777-88.
Thompson, S.K. (1990) Adaptive cluster sampling. Journal of the American Statistical Association, 85, 1050-59.
Thompson, S.K. and Seber, G.A.F. (1996) Adaptive sampling, John Wiley, New York.
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Mohammad, S.M. Rao-Blackwell versions of the Horvitz-Thompson and Hansen-Hurwitz in adaptive cluster sampling. Environmental and Ecological Statistics 6, 183–195 (1999). https://doi.org/10.1023/A:1009670205509
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DOI: https://doi.org/10.1023/A:1009670205509