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Comparison of variance estimators for the ratio estimator based on small sample

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Abstract

This paper sheds light on an open problem put forward by Cochran[1]. The comparison between two commonly used variance estimators\(v_1 (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{R} )\) and\(v_2 (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{R} )\) of the ratio estimator\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{R} \) for population ratioR from small sample selected by simple random sampling is made following the idea of the estimated loss approach (See [2]). Considering the superpopulation model under which the ratio estimator\(\hat \bar Y_R \) for population mean\(\overline Y \) is the best linear unbiased one, the necessary and sufficient conditions for\(\upsilon _1 (\hat R)\mathop \succ \limits^u \upsilon _2 (\hat R)\) and\(\upsilon _2 (\hat R)\mathop \succ \limits^u \upsilon _1 (\hat R)\) are obtained with ignored the sampling fractionf. For a substantialf, several rigorous sufficient conditions for\(\upsilon _2 (\hat R)\mathop \succ \limits^u \upsilon _1 (\hat R)\) are derived.

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References

  1. W.G. Cochran. Sampling Techniques. 3rd ed., John Wiley, New York, 1977

    MATH  Google Scholar 

  2. K.L. Lu, J.O. Berger. Estimation of Means: the Estimated Loss Approach. In: Sino-American Statistical Meeting Contributed Papers, 1987, 20–22

  3. W.G. Cochran Sampling Techniques, 2rd ed., John Wiley, New York, 1963

    Google Scholar 

  4. P.V. Sukhatme, B.V. Sukhatme. Sampling Theory of Surveys with Applications. 2nd ed., Asia Publishing House, London, 1970

    MATH  Google Scholar 

  5. L. Kish. Survey Sampling. John Wiley, New York, 1965

    MATH  Google Scholar 

  6. M.N. Murthy. Sampling Theory and Methods. Statistical Publishing Society, Calcutta, India, 1967

    MATH  Google Scholar 

  7. S. Feng, G. Zou. Small Sample Comparison of the Two Usual Variance Estimators of the Ratio Estimator.Acta Mathematica Sinica, 1996, 39: 764–776 (in Chinese)

    MathSciNet  Google Scholar 

  8. S. Feng, G. Zou. Comparison of Variance Estimators under Systematic Sampling Design.J. Sys. Sci. and Math. Sci., 1998, 18: 47–57 (in Chinese)

    MATH  MathSciNet  Google Scholar 

  9. R. Sundberg. Precision Estimation in Sample Survey Inference: A Criterion for Choice Between Variance Estimators.Biometrika, 1994, 81: 157–172

    Article  MATH  MathSciNet  Google Scholar 

  10. G. Zou, H. Liang. Admissibility of the Usual Estimators under Error-in-variables Superpopulation model.Statistics and Probability Letters, 1997, 32: 301–309

    Article  MATH  MathSciNet  Google Scholar 

  11. K.R.W. Brewer. Ratio Estimation in Finite Populations: Some Results Deducible from the Assumption of an Underlying Stochastic Process.Aust. J. Statist., 1963, 5: 93–105

    Article  MATH  MathSciNet  Google Scholar 

  12. R.M. Royall. On Finite Population Sampling Theory under Certain Linear Regression Models.Biometrika, 1970, 57: 377–387

    Article  MATH  Google Scholar 

  13. J. Durbin. A Note on the Application of Quenouille's Method of Bias Reduction to the Estimation of Ratios.Biometrika, 1959, 46: 477–480

    MATH  MathSciNet  Google Scholar 

  14. J.N.K. Rao, J.T. Webster. On two Methods of Bias Reduction in the Estimation of Ratios.Biometrika, 1966, 53: 571–577

    MATH  MathSciNet  Google Scholar 

  15. P.S.R.S. Rao. On Three Procedures of Sampling from Finite Populations.Biometrika, 1968, 55: 438–441

    Article  Google Scholar 

  16. P.S.R.S. Rao, J.N.K. Rao. Small Sample Results for Ratio Estimators.Biometrika, 1971, 58: 625–630

    Article  MATH  Google Scholar 

  17. C.M. Cassel, C.E. Särndal, J.H. Wretman. Some Results on Generalized Difference Estimation and Generalized Regression Estimation for Finite Populations.Biometrika, 1976, 63: 615–620

    Article  MATH  MathSciNet  Google Scholar 

  18. C.F.J. Wu. Estimation of Variance of the Ratio Estimator.Biometrika, 1982, 69: 183–189

    Article  MathSciNet  Google Scholar 

  19. J.N.K. Rao, R.R. Sitter. Variance Estimation under Two-phase Sampling with Application to Imputation for Missing Data.Biometrika, 1995, 82: 453–460

    Article  MATH  MathSciNet  Google Scholar 

  20. P.L. Hsu. On the Best Unbiased Quadratic Estimate of the Variance.Statist. Res. Mem., 1938, 2: 91–104

    Google Scholar 

  21. C.E. Särndal. Methods for Estimating the Precision of Survey Estimates when Imputation has been Used.Survey Methodology, 1992, 18: 241–252

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Beijing Normal University, 100875, Beijing, China

    Qin Huaizhen

  2. Institute of System Sciences, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, 100080, Beijing, China

    Li Lili

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  1. Qin Huaizhen
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  2. Li Lili
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This research is supported by the National Natural Science Foundation of China (No. 10071091).

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Huaizhen, Q., Lili, L. Comparison of variance estimators for the ratio estimator based on small sample. Acta Mathematicae Applicatae Sinica 17, 449–456 (2001). https://doi.org/10.1007/BF02669697

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  • DOI: https://doi.org/10.1007/BF02669697

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