On Some Useful “Inefficient” Statistics

  • Frederick Mosteller
Part of the Springer Series in Statistics book series (SSS)

Summary

Several statistical techniques are proposed for economically analyzing large masses of data by means of punched-card equipment; most of these techniques require only a counting sorter. The methods proposed are designed especially for situations where data are inexpensive compared to the cost of analysis by means of statistically “efficient”or “most powerful” procedures. The principal technique is the use of functions of order statistics, which we call systematic statistics.

It is demonstrated that certain order statistics are asymptotically jointly distributed according to the normal multivariate law.

For large samples drawn from normally distributed variables we describe and give the efficiencies of rapid methods:
  1. i)

    for estimating the mean by using 1, 2,⋯, 10 suitably chosen order statistics; (cf. p. 86)

     
  2. ii)

    for estimating the standard deviation by using 2, 4, or 8 suitably chosen order statistics; (cf. p. 89)

     
  3. iii)

    for estimating the correlation coefficient whether other parameters of the normal bivariate distribution are known or not (three sorting and three counting operations are involved) (cf. p. 94).

     

The efficiencies of procedures ii) and iii) are compared with the efficiencies of other estimates which do not involve sums of squares or products.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    G. W. Brown and J. W. Tukey, “Some distributions of sample means,” Annals of Math. Stat., Vol. 17 (1946), p. 1.MathSciNetGoogle Scholar
  2. [2]
    J. H. Curtiss, “A note on the theory of moment generating functions,” Annals of Math. Stat., Vol. 13 (1942), p. 430.MathSciNetGoogle Scholar
  3. [3]
    R. A. Fisher and L. H. C. Tippett, “Limiting forms of the frequency distribution of the largest or smallest member of a sample,” Proc. Camb. Phil. Soc., Vol. 24 (1928), p. 180.Google Scholar
  4. [4]
    R. A. Fisher and F. Yates, Statistical Tables, Oliver and Boyd, London, 1943.MATHGoogle Scholar
  5. [5]
    E. J. Gumbel, “Les valeurs extrêmes des distribution statistiques,” Annales de l’Institute Henri Poincaré, Vol. 4 (1935), p. 115.MathSciNetGoogle Scholar
  6. [6]
    E. J. Gumbel, “Statische Theorie der grossten Werte,” Zeitschrift für schweizerische Statistik und Volkswirtschaft, Vol. 75, part 2 (1939), p. 250.Google Scholar
  7. [7]
    H. O. Hartley, “The probability integral of the range in samples of n observations from a normal population,” Biometrika, Vol. 32 (1942), p. 301.MathSciNetGoogle Scholar
  8. [8]
    D. Jackson, “Note on the median of a set of numbers,” Bull. Amer. Math. Soc., Vol. 27 (1921), p. 160.Google Scholar
  9. [9]
    T. Kelley, “The selection of upper and lower groups for the validation of test items,” Jour. Educ. Psych., Vol. 30 (1939), p. 17.CrossRefGoogle Scholar
  10. [10]
    M. G. Kendall, The Advanced Theory of Statistics, J. B. Lippincott Co., 1943.Google Scholar
  11. [11]
    J. F. Kenney, Mathematics of Statistics, Part II, D. Van Nostrand Co., Inc., 1939, Chap. VII.Google Scholar
  12. [12]
    K. R. Nair and M. P. Shrivastava, “On a simple method of curve fitting,” Sankhya, Vol. 6, part 2 (1942), p. 121.MathSciNetGoogle Scholar
  13. [13]
    K. Pearson, “On the probable errors of frequency constants, Part III,” Biometrika, Vol. 13 (1920), p. 113.Google Scholar
  14. [14]
    K. Pearson (Editor), Tables for Statisticians and Biometricians, Part II, 1931, p. 78, Table VIII.Google Scholar
  15. [15]
    N. Smirnoff, “Über die Verteilung des allgemeinen Gliedes in der Variationsreihe,” Metron, Vol. 12 (1935), p. 59.Google Scholar
  16. [16]
    N. Smirnoff, “Sur la dependance des membres d’un series de variations,” Bull. Univ. État Moscou, Series Int., Sect. A., Math. et Mécan., Vol. 1, fasc.4, p. 1.Google Scholar
  17. [17]
    L. H. C. Tippett, “On the extreme individuals and the range of samples taken from a normal population,” Biometrika, Vol. 17 (1925), p. 364.Google Scholar
  18. [18]
    S. S. Wilks, Mathematical Statistics, Princeton University Press, Princeton, N. J., 1943.MATHGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  • Frederick Mosteller
    • 1
  1. 1.Princeton UniversityUSA

Personalised recommendations