What is direct and inverse variation

The direct and the reverse (inverse) conclusion in heterograde theory

Introduction to Mathematical Statistics pp 193-266 | Cite as


Before we begin our presentation, we believe we must again draw the reader's attention to the fact that the statistical aggregates with which the heterograde theory is confronted are incomparably more colorful and complicated than those which appear in the homograde theory. From the standpoint of the mathematician, the latter consist only of zeros and ones, which are mixed together in the proportion \ (\ frac {M} {N} ~ = ~ p ~ and ~ \ frac {NM} {N} = q \) (see above § 2, Chapter II), while the former can consist of any positive or negative whole or non-whole numbers, irrational numbers by no means excluded. From this it becomes evident that the propositions which express various general relationships between such totalities must be much more complicated on the one hand and much more indefinite on the other. If in the case of the homograde totalities one actually works with the quotients p and p'And all other statistical parameters that occur simply as functions of p, p′, N and n can represent, one has to do with the hetero-grade statistical totalities with an infinite series of statistical parameters: the series of mi, the μi, the product moments, the semi-invariants, the cumulants, etc. (see above Chapter II, § 5). And as long as no additional assumptions are made about the character of the distribution law of random variables, the exact binomial theorem (Chapter I, § 3) and the approximated exponential theorems by De Moivre-Laplace (Chapter I, § 4-6) and Poisson (Chapter I, § 12) only see Markoff adopting a generalized inequality (Chapter II, § 8), which, however, as we shall see later, allows a much greater range of variation for the probabilities in question and is significantly less advantageous in this respect . If one wants to specify the distribution law of the statistical totality more precisely, its variability has the effect that, depending on the concrete circumstances, a very large number of approximation formulas can be used and that, as it turns out, there is not a single one of them which in practice appears all cases would fit, because formulas that represent the law of distribution through an infinite series of statistical parameters can only be understood as "solutions" from the abstract mathematical and not from the practical-statistical point of view.

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This chapter is part of the Springer Book Archives digitization project with publications that have appeared since the publisher's inception in 1842. With this archive, the publisher provides sources for both historical and disciplinary research, which must be viewed in a historical context. This chapter is from a book that was published before 1945 and is therefore not advertised by the publisher in its political-ideological orientation typical of the time.


  1. Seimatsu Narumi: On further inequalities with possible application to problems in the theory of probability. Biometrika, Vol. XV, pp. 245-253, 1923. Google Scholar
  2. R. Frisch: Sur les semi-invariants et moments employés dans l’étude des distributions statistiques, Oslo 1926Google Scholar
  3. Det Norske Videnskabs-Akademis Skrifter, II, No. 3), and J. F. Steffen s e n: On the Sum or Integral of the Product of two Functions. Skandinavisk Aktuarietidskrift. Uppsala 1927. Google Scholar
  4. L. March: Analysis de la variabilité, Metron VI; further z. B. T. Jerneman: On the Substitution of constants in Sampling formulas with special reference to the theory of the too small values. Nordic Statistical Journal, Vol. 3, pp. 85-112. Google Scholar
  5. L. Isserlis: On the Moment Distributions of Moments in the Case of amples Drawn from a Limited Universe (Proceed. Of the Roy. Soc., A, Vol. 131, 1931), p. 587; Google Scholar
  6. See L. Is s e r l i s: On the Conditions under which the "Probable Errors" of Frequency Distributions have a real significance, Proceed. of the Roy. Soc., A, Vol. 92, pp. 23-41, 191Google Scholar
  7. Cf. L. Is s e r l i s: On the Conditions under which the "Probable Errors" of Frequency Distributions have a real significance, Proceed. of the Roy. Soc., A, Vol. LXXXI, Part I, pp. 75-81, January 1918Google Scholar
  8. G. Mortara: Elementi di Statistica, p. 356, Roma 1917; A. A. Tschuprow: On the theory of the stability of statistical series, p. 219Google Scholar
  9. S. S. Kohn: On the question of the application of the sampling method to the processing of agricultural surveys, Petrograd 1917 (official memorandum of the Ministry of Agriculture; Russian) Google Scholar
  10. M. Greenwood and L. Isserlis: A Historical Note on the Problem of Small Samples, Journ. of the Roy. statist. Soc., Vol. XC, pp. 347-352, 192 Google Scholar
  11. K. Pearson: Another "historical note on the problem of small samples", Biometrika, Vol XIX, pp. 207-210, 1927Google Scholar
  12. See also Tschuprow: On the Mathematical Expectation etc., Biometrika, Vol. XII, p. 192, Formula (3), 1919. Google Scholar
  13. See A. E. R. Church: On the Moments of the Distribution of squared standard Deviations for samples of N drawn from an indefinitely large Population, Biometrika, Vol. XVII, pp. 79-83, 1925Google Scholar
  14. Biometrika, Vol. XVIII, pp. 321-394, 1926Google Scholar
  15. same: Note on a Memoir by A. E. R. Church, Biometrika, Vol. XXIV, p. 292, 1932. Google Scholar
  16. On the question of determining the higher moments for a homograde ensemble see also: V. Romanovsky: Note on the Moments of a Binomial (p - {- q) n about its mean, Biometrika, Vol. XV, pp. 410-412, 1923Google Scholar
  17. K. Pearson: On the Moments of the hypergeometrical series, Biometrika, Vol. XVI, pp. 157-162, 1924Google Scholar
  18. V. Romanovsky: On the Moments of the hypergeo-metrical series, Biometrika, Vol. XVII, pp. 57-60, 1925Google Scholar
  19. Ragnar Frish: Recurrence formulas for the moments of the point binomial, Biometrika, Vol. XVII, pp. 165–171, 1925Google Scholar
  20. Biometrika, Vol. XXVI, pp. 262-264, 1934. Google Scholar
  21. E. S. Littlejohn: On an elementary method of finding the moments of the terms of a multiple hyper-geometrical series, Metron, Vol. I, No. 4, pp. 49-56, 1921 Google Scholar
  22. J. SplawaNeyman: Contribution to the theory of small samples drawn from a finite population, Biometrika, Vol. XVII, pp. 472-479, 1925Google Scholar
  23. V. Romanovsky: On the moments of standard deviations and of correlation coefficients in samples from normal population, Metron, Vol. V, No. 4, pp. 3–46, 1925Google Scholar
  24. Metron, Vol. VIII, Nos. 1 & 2, pp. 251-289, 1929Google Scholar
  25. Samuel S. Wilks: On the distribution of statistics in samples from a normal population of two variables with matched sampling of one variable, Metron, Vol. IX, No. 3 and 4, pp. 87-126, 1932Google Scholar
  26. William Dowell Baten: Frequency Laws for the Sum of n variables which are subject to given frequency laws, Metron, Vol. X, No. 3, pp. 75-91, 1932Google Scholar
  27. J. M. le Roux: A Study of the Distribution of the Variance in Small Samples, Biometrika, Vol. XXIII, pp. 134-190, 1931 CrossRefGoogle Scholar
  28. N. St. G e o r g e s c u: Further Contributions to the Sampling Problem, Biometrika, Vol. XXIV, pp. 65 to 107, 1932Google Scholar
  29. See F. Y. Edgeworth: The Law of Error, Cambridge Philosophical Transactions, XX (1904) Google Scholar
  30. See H. Bruns: Probability Calculation and Collective Measurement, pp. 39ff., Leipzig and Berlin 1906Google Scholar
  31. Best to look up at Mises: Probability Theory, pp. 265ff., Google Scholar
  32. See “Student”: The probable Error of a Mean, Biometrika, Vol. VI, pp. 1-25, 1908. Google Scholar
  33. See R. A. Fisher: Inverse probability, Proceed. Camb. Phil. Soc., Vol. XXVI, Part 4, 1930Google Scholar
  34. See Al. A. Tschuprow: Aims and ways of the stochastic foundation of the statistical theory. Nordisk Statistisk Tidskrift, Vol. 3, No. 4, p. 468, 1924. Google Scholar
  35. R. A. Fisher: A mathematical examination of determining the accuracy of an observation by the mean error and by the mean square error. Monthly Notices of the Royal Astronomical Society, LXXX, pp. 758-770 (1920). Google Scholar
  36. See N. P. Bertelsen: On the Compatibility of frequency Constants, and the presumtive laws of error, Skandinavisk Aktuarietidskrift, Uppsala 1927Google Scholar
  37. J. F. Steffensen: Some recent researches in the theory of statistics and actuarial science, Cambridge 1930 (Published for the Institute of Actuaries). Google Scholar
  38. A. A. Markoff: Probability Theory, chap. VII. Moscow 1924 (Russian) .Google Scholar
  39. Tschuprow: On the asymptotic frequency distribution of the arithmetic means of n correlated observations for very great values ​​of n: Journal of the Royal Statistical Society, Vol. LXXXVIII, pp. 91-104, 1925. Google Scholar
  40. E. C. Rho des: The Precision of Means and Standard Deviations when the Individual Errors are Correlated: Ibidem, Vol. XC, pp. 135-143, 1927Google Scholar
  41. On § 4. - Mrs. A. Willers: Estimation of distributions with upwardly concave cumulative curves: Journal for applied mathematics and mechanics, edited by R. Mises, Volume 13, Issue 5, 1933. Google Scholar
  42. On § 5. - J. Neyman: On the correlation of the mean and the variance in samples drawn from an “infinite” population: Biometrika, Vol. XVIII, pp. 401-413.Google Scholar
  43. On § 7. - Ragnar Frisch: On the use of difference equations in the study of frequency distributions: Metron, Vol. X, No. 3, pp. 35-59, 1932. Google Scholar
  44. L. v. Bortkiewicz: About a different error laws common property: meeting reports of the Berlin Mathematical Society, 1923.Google Scholar
  45. Karl Pearson: On the Mean-Error of Frequency Distributions: Biometrika, Vol. XVI, pp. 198-200.Google Scholar

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© Julius Springer in Vienna 1935

Authors and Affiliations

  1. 1. State University in SofiaSofiaBulgaria