Environmental and Ecological Statistics

, Volume 21, Issue 2, pp 285–311 | Cite as

Variances and variance estimators of the improved ratio estimators under adaptive cluster sampling



Although not design-unbiased, the ratio estimator is recognized as more efficient when a certain degree of correlation exists between the variable of primary interest and the auxiliary variable. Meanwhile, the Rao–Blackwell method is another commonly used procedure to improve estimation efficiency. Various improved ratio estimators under adaptive cluster sampling (ACS) that make use of the auxiliary information together with the Rao–Blackwellized univariate estimators have been proposed in past research studies. In this article, the variances and the associated variance estimators of these improved ratio estimators are proposed for a thorough framework of statistical inference under ACS. Performance of the proposed variance estimators is evaluated in terms of the absolute relative percentage bias and the empirical mean-squared error. As expected, results show that both the absolute relative percentage bias and the empirical mean-squared error decrease as the initial sample size increases for all the variance estimators. To evaluate the confidence intervals based on these variance estimators and the finite-population Central Limit Theorem, the coverage rate and the interval width are used. These confidence intervals suffer a disadvantage similar to that of the conventional ratio estimator. Hence, alternative confidence intervals based on a certain type of adjusted variance estimators are constructed and assessed in this article.


Adaptive cluster sampling Confidence interval Minimal sufficient statistic Ratio estimator Sufficient statistic Variance estimator 



Support for this research is provided by the National Science Council, Taiwan, NSC 100-2118-M-006-004. The authors are grateful to the editor and referees for their insightful and constructive comments/suggestions that greatly improved the presentation and quality of the article.


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Statistics, School of ManagementNational Cheng-Kung UniversityTainanTaiwan

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