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

- 28 Downloads
- 3 Citations

## Summary

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.

## Keywords

Sampling Design Statistical Inference Sample Survey Finite Population Distinct Sample## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Fisher, R. A. (1935).
*The Design of Experiments*, Oliver and Boyd, Edinburgh.Google Scholar - [2]Bowley, A. L. (1926). Measurement of the precision attained in sampling,
*Bull. Int. Statist. Inst.*,**22**, 6–62.Google Scholar - [3]Mahalanobis, P. C. (1944). On large-scale sample surveys,
*Phil. Trans. R. Soc., B*,**231**, 329–451.CrossRefGoogle Scholar - [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]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]Yates, F. (1946). A review of recent statistical developments in sampling and sampling surveys,
*J. R. Statist. Soc.*,**109**, 12–43.CrossRefGoogle Scholar - [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]Cassel, C.-M., Särndal, C.-E. and Wretman, J. H. (1977).
*Foundations of Inference in Survey Sampling*, Wiley, New York.zbMATHGoogle Scholar - [9]Midzuno, H. (1950). An outline of the theory of sampling systems,
*Ann. Inst. Statist. Math.*,**1**, 149–156.MathSciNetCrossRefGoogle Scholar - [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]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]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]Godambe, V. P. (1966). A new approach to sampling from finite populations I,
*J. R. Statist. Soc.*, B,**28**, 310–319.zbMATHGoogle Scholar - [14]Godambe, V. P. (1969). A fiducial argument with applications in survey sampling,
*J. R. Statist. Soc.*, B,**31**, 240–260.Google Scholar - [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]Bowley, A. L. (1920).
*Elements of Statistics*, King, London.zbMATHGoogle Scholar - [17]United Nations (1948).
*Recommendations for the Preparation of Sample Survey Reports*, Series C, No. 1, United Nations, New York.Google Scholar - [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]Hájek, J. (1959). Optimum strategy and other problems in probability sampling,
*Câs. pêst. Math.*,**84**, 387–423.MathSciNetzbMATHGoogle Scholar - [20]Godambe, V. P. (1960). An admissible estimate for any sampling design,
*Sankhyā*,**22**, 285–288.MathSciNetzbMATHGoogle Scholar - [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]Rao, C. R. (1975). Some problems in sample surveys,
*Suppl. Adv. Appl. Prob.*,**7**, 50–61.CrossRefGoogle Scholar - [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]Goodman, R. and Kish, L. (1950). Controlled selection—a technique in probability sampling,
*J. Amer. Statist. Ass.*,**45**, 350–372.Google Scholar - [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]Jessen, R. J. (1970). Probability sampling with marginal constraints,
*J. Amer. Statist. Ass.*,**65**, 776–796.CrossRefGoogle Scholar - [27]Smith, H. F. (1955). Variance components, finite populations, and experimental inference,
*N. C. Inst. Statistics Mimeo Series*, 135.Google Scholar - [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]Fisher, R. A. (1956).
*Statistical Methods and Scientific Inference*, Oliver and Boyd, Edinburgh.zbMATHGoogle Scholar - [30]Fisher, R. A. (1959). Mathematical probability in the natural sciences,
*Metrika*,**2**, 1–10.MathSciNetCrossRefGoogle Scholar - [31]Pitman, E. J. G. (1937). The “closest” estimates of statistical parameters,
*Proc. Camb. Phil. Soc.*,**33**, 212–222.CrossRefGoogle Scholar - [32]Pearson, K. (1919). On the generalized Tchebycheff theorem in the mathematical theory of statistics,
*Biometrika*,**12**, 284–296.CrossRefGoogle Scholar - [33]Irwin, J. O. and Kendall, M. G. (1944). Sampling moments for a finite population,
*Ann. Eugenics*,**12**, 138–142.MathSciNetCrossRefGoogle Scholar - [34]Tukey, J. W. (1950). Some sampling simplified,
*J. Amer. Statist. Ass.*,**45**, 501–519.MathSciNetCrossRefGoogle Scholar - [35]Wishart, J. (1952). Moment coefficients of the
*k*-statistics in samples from a finite population,*Biometrika*,**39**, 1–13.MathSciNetCrossRefGoogle Scholar - [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 in
*N. C. Inst. Statistics Mimeo Series*, 296, 1961.)Google Scholar - [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]
- [39]Basu, D. (1969). Role of sufficiency and likelihood principles in sample survey theory,
*Sankhyā*, A,**31**, 441–454.MathSciNetzbMATHGoogle Scholar - [40]Cochran, W. G. (1963).
*Sampling Techniques*, Wiley, New York.zbMATHGoogle Scholar - [41]Hastings, W. K. (1974). Variance reduction and nonnormality,
*Biometrika*,**61**, 143–149.MathSciNetCrossRefGoogle Scholar