Probability Theory and Related Fields

, Volume 70, Issue 1, pp 131–156 | Cite as

Local asymptotic normality of sampling experiments

  • Hartmut Milbrodt


Following Hájek's and Madow's asymptotic approach to classical survey sampling, a framework for the asymptotic analysis of superpopulation models is proposed. Within this framework weak limit theorems for sequences of experiments obtained by Poisson sampling and Rejective sampling are derived. In particular, it is shown that the sampling experiments are locally asymptotically normal in LeCam's sense, if the underlying superpopulation model is an L2-generated regression experiment. This result can e.g. be used to produce Horvitz-Thompson-type estimators which are asymptotically efficient median-unbiased estimates of certain functionals of the regression function.


Sampling Experiment Sampling Plan Inclusion Probability Triangular Array Rejective Sampling 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Arnab, R.: An Addendum to Singh and Singh's Paper on Random Non-Response in Unequal Probability Sampling. Sankhya, Ser. C, 41, 138–140 (1979)MATHGoogle Scholar
  2. 2.
    Becker, C.: Schwache asymptotische Normalität von statistischen Experimenten bei unabhängigen, nicht notwendig identisch verteilten Beobachtungen. Bayreuther Math. Schr. 13, 1–153 (1983)MATHGoogle Scholar
  3. 3.
    Cassel, C., Särndal, C., Wretman, J.H.: Foundations of Inference in Survey Sampling. New York: Wiley 1977MATHGoogle Scholar
  4. 4.
    Cochran, W.G.: Sampling Techniques (3rd ed.) New York: Wiley 1977MATHGoogle Scholar
  5. 5.
    Freedman, D.: A Remark on the Difference between Sampling with and without Replacement. J. Am. Stat. Assoc. 73, 681 (1977)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hájek, J.: Asymptotic theory of rejective sampling with varying probabilities from a finite population. Ann. Math. Stat. 35, 1491–1523 (1964)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hájek, J.: Sampling from a finite population. New York: Dekker 1981MATHGoogle Scholar
  8. 8.
    Ibragimov, I.A., Has'minskii, R.Z.: Statistical Estimation. New York-Heidelberg-Berlin: Springer 1981CrossRefMATHGoogle Scholar
  9. 9.
    Kallenberg, O.: A note on the asymptotic equivalence of sampling with and without replacement. Ann. Stat. 2, 819–821 (1974)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    LeCam, L.: Sufficiency and approximate sufficiency. Ann. Math. Stat. 35, 1419–1455 (1964)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    LeCam, L.: Notes on asymptotic methods in statistical decision theory. I. Publ. du Centre de Recherches Math., Univ. de Montréal (1974)Google Scholar
  12. 12.
    Madow, W.G.: On the limiting distributions of estimates based on samples from finite universes. Ann. Math. Stat. 19, 535–545 (1948)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Milbrodt, H.: Limits of Experiments associated with Sampling Plans. Preprint, Bayreuth (1985)Google Scholar
  14. 14.
    Milbrodt, H., Strasser, H.: Limits of Triangular Arrays of Experiments. In: Infinitely Divisible Statistical Experiments, pp. 14–54. Lecture Notes Stat. 27. Berlin-Heidelberg-New York: Springer 1985Google Scholar
  15. 15.
    Millar, P.W.: The Minimax Principle in Asymptotic Statistical Theory. In: Lecture Notes Math. 976, pp. 76–265. Berlin-Heidelberg-New York: Springer 1982Google Scholar
  16. 16.
    Moussatat, M.W.: On the Asymptotic Theory of Statistical Experiments and some of its Applications. Ph. D. Dissertation, Berkeley (1976)Google Scholar
  17. 17.
    Rao, J.N.K., Scott, A.J.: The Analysis of Categorical Data from Complex Sample Surveys: Chi-Squared Tests for Goodness of Fit and Independence in Two-Way Tables. J. Am. Stat. Assoc. 76, 221–230 (1981)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Särndal, C.E.: Two model-based inference arguments in Survey Sampling. Aust. J. Stat. 22, 341–348 (1980)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Strasser, H.: Mathematical Theory of Statistics. Statistical Experiments and Asymptotic Decision Theory. Berlin: de Gruyter 1985 (to appear)CrossRefMATHGoogle Scholar
  20. 20.
    Thomson, I.: Design and Estimation Problems when Estimating a Regression Coefficient from Survey Data. Metrika 25, 27–35 (1978)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Torgersen, E.N.: Comparison of statistical experiments. Scand. J. Stat. 3, 186–208 (1976)MathSciNetMATHGoogle Scholar
  22. 22.
    Torgersen, E.N.: Comparison of some statistical experiments associated with sampling plans. Prob. Math. Stat. 3, 1–17 (1982)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Hartmut Milbrodt
    • 1
  1. 1.Mathematik VII der Universität BayreuthBayreuthFederal Republic of Germany

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