On statistical inference in sample surveys and the underlying role of randomization

  • J. C. Koop


Assuming only the existence of a universe and a frame that identifies its members (ultimate sampling units), and a minimum number of necessary but nonrestrictive assumptions, the writer derives a basic proposition which shows that for sample surveys any form of inference about any universe characteristic must depend on the sampling distribution of estimates generated by randomization, and, by direct implication, the sampling design. Unbiasedness is validated by this proposition, but likelihood appears to be in conflict with it, and by implication with randomization. The relation between high or low variance and correspondingly low or high probability for an estimator is also investigated in the paper. Finally it is argued that restrictions on randomization may be injurious to normality.


Sampling Design Statistical Inference Sample Survey Finite Population Distinct Sample 


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  1. [1]
    Fisher, R. A. (1935).The Design of Experiments, Oliver and Boyd, Edinburgh.Google Scholar
  2. [2]
    Bowley, A. L. (1926). Measurement of the precision attained in sampling,Bull. Int. Statist. Inst.,22, 6–62.Google Scholar
  3. [3]
    Mahalanobis, P. C. (1944). On large-scale sample surveys,Phil. Trans. R. Soc., B,231, 329–451.CrossRefGoogle Scholar
  4. [4]
    Neyman, J. (1934). On the two different aspects of the representative method: the method of stratified sampling and the method of purposive selection,J. R. Statist. Soc.,97, 588–625.CrossRefGoogle Scholar
  5. [5]
    Gini, C. and Galvani L. (1929). Di una applicazione del metoda rappresentativo all' ultimo censimento italiano della populazione,Annali di Statistica,4, 1–107.Google Scholar
  6. [6]
    Yates, F. (1946). A review of recent statistical developments in sampling and sampling surveys,J. R. Statist. Soc.,109, 12–43.CrossRefGoogle Scholar
  7. [7]
    Kempthorne, O. (1973). Inference from experiments and randomization,A Survey of Statistical Design and Linear Models (ed. J. N. Srivastava), North Holland, Amersterdam, 303–331.Google Scholar
  8. [8]
    Cassel, C.-M., Särndal, C.-E. and Wretman, J. H. (1977).Foundations of Inference in Survey Sampling, Wiley, New York.zbMATHGoogle Scholar
  9. [9]
    Midzuno, H. (1950). An outline of the theory of sampling systems,Ann. Inst. Statist. Math.,1, 149–156.MathSciNetCrossRefGoogle Scholar
  10. [10]
    Midzuno, H. (1952). On the sampling system with probability proportionate to the sum of sizes,Ann. Inst. Statist. Math.,3, 99–107.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Horvitz, D. G. and Thompson, D. J. (1952). A generalization of sampling without replacement from a finite universe,J. Amer. Statist. Ass.,47, 663–685.MathSciNetCrossRefGoogle Scholar
  12. [12]
    Koop, J. C. (1963). On the axioms of sample formation and their bearing on the construction of linear estimators in sampling theory for finite universes I, II, III,Metrika,7, 81–114 and 165–204.MathSciNetCrossRefGoogle Scholar
  13. [13]
    Godambe, V. P. (1966). A new approach to sampling from finite populations I,J. R. Statist. Soc., B,28, 310–319.zbMATHGoogle Scholar
  14. [14]
    Godambe, V. P. (1969). A fiducial argument with applications in survey sampling,J. R. Statist. Soc., B,31, 240–260.Google Scholar
  15. [15]
    Koop, J. C. (1974). Notes for a unified theory of estimation for sample surveys taking into account response errors,Metrika,21, 19–39.MathSciNetCrossRefGoogle Scholar
  16. [16]
    Bowley, A. L. (1920).Elements of Statistics, King, London.zbMATHGoogle Scholar
  17. [17]
    United Nations (1948).Recommendations for the Preparation of Sample Survey Reports, Series C, No. 1, United Nations, New York.Google Scholar
  18. [18]
    Yamamoto, S. (1955). On the theory of sampling with probabilities proportionate to given values,Ann. Inst. Statist. Math.,7, 25–38.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Hájek, J. (1959). Optimum strategy and other problems in probability sampling,Câs. pêst. Math.,84, 387–423.MathSciNetzbMATHGoogle Scholar
  20. [20]
    Godambe, V. P. (1960). An admissible estimate for any sampling design,Sankhyā,22, 285–288.MathSciNetzbMATHGoogle Scholar
  21. [21]
    Basu, D. (1971). An essay on the logical foundations of survey sampling, part one,Foundations of Statistical Inference (ed. V. P. Godambe and D. A. Sprott), Holt, Rinehart and Winston, Toronto, 203–242.Google Scholar
  22. [22]
    Rao, C. R. (1975). Some problems in sample surveys,Suppl. Adv. Appl. Prob.,7, 50–61.CrossRefGoogle Scholar
  23. [23]
    Hájek, J. (1964). Asymptotic theory of rejective sampling with varying probabilities from a finite population,Ann. Math. Statist.,35, 1491–1523.MathSciNetCrossRefGoogle Scholar
  24. [24]
    Goodman, R. and Kish, L. (1950). Controlled selection—a technique in probability sampling,J. Amer. Statist. Ass.,45, 350–372.Google Scholar
  25. [25]
    Sukhatme, B. V. and Avadhani, M. S. (1965). Controlled selection a technique in random sampling,Ann. Inst. Statist. Math.,17, 15–28.MathSciNetCrossRefGoogle Scholar
  26. [26]
    Jessen, R. J. (1970). Probability sampling with marginal constraints,J. Amer. Statist. Ass.,65, 776–796.CrossRefGoogle Scholar
  27. [27]
    Smith, H. F. (1955). Variance components, finite populations, and experimental inference,N. C. Inst. Statistics Mimeo Series, 135.Google Scholar
  28. [28]
    Barnard, G. A., Jenkins, G. M. and Winsten, C. B. (1962). Likelihood inference and time series (with discussion),J. R. Statist. Soc., A,125, 321–372.CrossRefGoogle Scholar
  29. [29]
    Fisher, R. A. (1956).Statistical Methods and Scientific Inference, Oliver and Boyd, Edinburgh.zbMATHGoogle Scholar
  30. [30]
    Fisher, R. A. (1959). Mathematical probability in the natural sciences,Metrika,2, 1–10.MathSciNetCrossRefGoogle Scholar
  31. [31]
    Pitman, E. J. G. (1937). The “closest” estimates of statistical parameters,Proc. Camb. Phil. Soc.,33, 212–222.CrossRefGoogle Scholar
  32. [32]
    Pearson, K. (1919). On the generalized Tchebycheff theorem in the mathematical theory of statistics,Biometrika,12, 284–296.CrossRefGoogle Scholar
  33. [33]
    Irwin, J. O. and Kendall, M. G. (1944). Sampling moments for a finite population,Ann. Eugenics,12, 138–142.MathSciNetCrossRefGoogle Scholar
  34. [34]
    Tukey, J. W. (1950). Some sampling simplified,J. Amer. Statist. Ass.,45, 501–519.MathSciNetCrossRefGoogle Scholar
  35. [35]
    Wishart, J. (1952). Moment coefficients of thek-statistics in samples from a finite population,Biometrika,39, 1–13.MathSciNetCrossRefGoogle Scholar
  36. [36]
    Koop, J. C. (1957). Contributions to the general theory of sampling finite populations without replacement and with unequal probabilities, Ph.D. Thesis, N. C. State Univ. (Also inN. C. Inst. Statistics Mimeo Series, 296, 1961.)Google Scholar
  37. [37]
    Godambe, V. P. and Joshi, V. M. (1965). Admissibility and Bayes estimation in sampling from finite populations: I,Ann. Math. Statist.,36, 1707–1722.MathSciNetCrossRefGoogle Scholar
  38. [38]
    Godambe, V. P. (1975). A reply to my critics,Sankhyā, C,37, 53–76.zbMATHGoogle Scholar
  39. [39]
    Basu, D. (1969). Role of sufficiency and likelihood principles in sample survey theory,Sankhyā, A,31, 441–454.MathSciNetzbMATHGoogle Scholar
  40. [40]
    Cochran, W. G. (1963).Sampling Techniques, Wiley, New York.zbMATHGoogle Scholar
  41. [41]
    Hastings, W. K. (1974). Variance reduction and nonnormality,Biometrika,61, 143–149.MathSciNetCrossRefGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 1979

Authors and Affiliations

  • J. C. Koop
    • 1
  1. 1.Research Traingle Institute

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