Skip to main content
SpringerLink
Log in

The Impact of Wassily Hoeffding’s Research on Nonparametrics

  • Chapter
The Collected Works of Wassily Hoeffding

Part of the book series: Springer Series in Statistics ((PSS))

Abstract

Wassily Hoeffding earned his Ph.D. degree in mathematics from the Berlin University in 1940 for a dissertation in correlation theory which dealt with some aspects of bivariate probability distributions that are invariant under monotone transformations of the marginals. This dissertation was primarily devoted to some (descriptive) studies of certain measures of rank correlations. With the impending Second World War, for Wassily, living in Berlin in the early forties was not that comfortable. Nevertheless, he managed to advance his basic research on nonparametric correlation theory. It was only after his eventual migration to the United States (in the Fall of 1946) that he started to appreciate the full depth of probability theory and statistics (during his sojourn at the Columbia University, New York), and most of his pioneering work emerged during his longtime residence at Chapel Hill (1947–1991). He felt that “…probability and statistics were very poorly represented in Berlin at that time (1936–1945) …”. Notwithstanding this, his early work on correlation theory was not just a landmark in nonparametrics; it also endowed him with a career-long zeal and affection for the pursuit of the most fundamental research in mathematical statistics, probability theory, numerical analysis and a variety of other related areas. In this respect, nonparametrics was indisputedly the “jewel in the crown” of Wassily’s creativity and ingenuity in research. Wassily Hoeffding indeed played a seminal role in stimulating basic research in a broad domain of mathematical statistics and probability theory, and his “collected work” in this volume reflects the genuine depth and immense breadth of his research contributions. In this article, I shall mainly confine myself to describing the profound impact of his work in the general area of nonparametrics, with occasional detours to some other related areas.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson, T.W. and Samuels, S.M. (1966). Some inequalities among binomial and Poisson probabilities. Proc. Fifth Berkeley Symp. Math. Statist. Probab. I, 1–12.

    Google Scholar 

  2. Bahadur, R.R. (1960). Asymptotic efficiency of tests and estimates. Sankhya 22, 229–252.

    MathSciNet  MATH  Google Scholar 

  3. Berk, R.H. (1966). Limiting behavior of posterior distributions when the model is incorrect. Ann. Math. Statist., 37, 51–58.

    Article  MathSciNet  MATH  Google Scholar 

  4. Bickel, P.J. and van Zwet, W.R. (1980). On a theorem of Hoeffding. In Asymptotic Theory of Statistical Tests and Estimation (ed: I.M. Chakravarti), Academic Press, New York, 307–324.

    Google Scholar 

  5. Chernoff, H. and Savage, I.R. (1958). Asymptotic normality and efficiency of certain nonparametric test statistic. Ann. Math. Statist. 29, 972–994.

    Article  MathSciNet  Google Scholar 

  6. Dwass, M. (1953). On the asymptotic normality of certain rank order statistics. Ann. Math. Statist. 24, 303–306.

    Article  MathSciNet  MATH  Google Scholar 

  7. Dwass, M. (1955). On the asymptotic normality of some statistics used in nonparametric tests. Ann. Math. Statist. 26, 334–339.

    Article  MathSciNet  MATH  Google Scholar 

  8. Dwass, M. (1957). On the distribution of ranks and certain rank order statistics. Ann. Math. Statist. 28, 424–431.

    Article  MathSciNet  MATH  Google Scholar 

  9. Fernholz, L.T. (1983). Von Mises Calculus for Statistical Functionals. Lecture Notes in Statistics, No.19, Springer-Verlag, New York.

    Book  Google Scholar 

  10. Fisher, R.A. (1935). The logic of inductive inference. J. B. Statist. Soc. 98, 39–82.

    Article  Google Scholar 

  11. Gleser, L.J. (1975). On the distribution of the number of successes in independent trials. Ann. Probab. 3, 182–188.

    Article  MathSciNet  MATH  Google Scholar 

  12. Govindarajulu, Z., LeCam, L. and Raghavachari, M. (1966). Generalizations of theorems of Chernoff and Savage on asymptotic normality of nonparametric test statistics. In Proc. Fifth Berkeley Symp. Math. Statist. Probab. I, 609–638.

    Google Scholar 

  13. Hájek, J. (1961). Some extensions of the Wald-Wolfowitz-Noether theorem. Ann. Math. Statist. 32, 506–523.

    Article  MathSciNet  MATH  Google Scholar 

  14. Hájek, J. (1962). Asymptotically most powerful rank order tests. Ann. Math. Statist. 33, 1124–1147.

    Article  MathSciNet  MATH  Google Scholar 

  15. Hájek, J. (1968). Asymptotic normality of simple linear rank statistics under alternatives. Ann. Math. Statist. 39, 325–346.

    Article  MathSciNet  MATH  Google Scholar 

  16. Hájek, J. and Sidak, Z. (1967). Theory of Rank Tests. Academic Press, Prague.

    Google Scholar 

  17. Halmos, P.R. (1946). The theory of unbiased estimation. Ann. Math. Statist. 17, 34–44.

    Article  MathSciNet  MATH  Google Scholar 

  18. Herr, D.G. (1967). Asymptotically optimal tests for multivariate normal distributions. Ann. Math. Statist. 38, 1829–1844.

    Article  MathSciNet  MATH  Google Scholar 

  19. Hoeffding, W. (1940). Masstabinvariante Korrelationstheorie. Schriften des Mathematischen Instituts und des Instituts für Angewandte Mathematik der Universität Berlin. 5, Heft 3, 179–233.

    Google Scholar 

  20. Hoeffding, W. (1941). Masstabinvariante Korrelationsmasse für diskontinuierliche Verteilungen. Archiv für mathematischen Wirtschaften und Sozialforschung. 7, 49–70.

    Google Scholar 

  21. Hoeffding, W. (1942) Stochastische abhängigkeit und funktionaler Zusammenhang. Skand. Aktuar. 25, 200–227.

    Google Scholar 

  22. Hoeffding, W. (1948) A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19, 293–325.

    Article  MathSciNet  MATH  Google Scholar 

  23. Hoeffding, W. and Robbins, H. (1948) The central limit theorems for dependent random variables. Duke Math. J. 15, 773–780.

    Article  MathSciNet  MATH  Google Scholar 

  24. Hoeffding, W. (1948) A nonparametric test of independence. Ann. Math. Statist. 19, 546–547.

    Article  MathSciNet  MATH  Google Scholar 

  25. Hoeffding, W. (1951) ‘Optimum’ nonparametric tests. Proc. Second Berkeley Symp. Math. Statist. Probab., University of California Press, 83-92.

    Google Scholar 

  26. Hoeffding, W. (1951) A combinatorial central limit theorem. Ann. Math. Statist. 22, 558–566.

    Article  MathSciNet  MATH  Google Scholar 

  27. Hoeffding, W. (1952) The large-sample power of tests based on permutations of observations. Ann. Math. Statist. 23, 169–192.

    Article  MathSciNet  MATH  Google Scholar 

  28. Hoeffding, W. (1953) On the distribution of the expected values of order statistics. Ann. Math. Statist. 24, 93–100.

    Article  MathSciNet  MATH  Google Scholar 

  29. Hoeffding, W. and Rosenblatt, J.R. (1955) The efficiency of tests. Ann. Math. Statist. 26, 52–63.

    Article  MathSciNet  MATH  Google Scholar 

  30. Hoeffding, W. (1956) On the distribution of the number of successes in independent trials. Ann. Math. Statist. 27, 713–721.

    Article  MathSciNet  MATH  Google Scholar 

  31. Hoeffding, W. (1961) The Strong Law of Large Numbers for U-Statistics. Inst. Statist. UNC Mimeo Report No. 302.

    Google Scholar 

  32. Hoeffding, W. (1963) Probability Inequalities for sums of bounded random variables. J. Am. Statist. Assoc. 58, 13–30.

    Article  MathSciNet  MATH  Google Scholar 

  33. Hoeffding, W. (1965) Asymptotically optimal tests for multinomial distributions (with Discussion). Ann. Math. Statist. 36 369–408.

    Article  MathSciNet  MATH  Google Scholar 

  34. Hoeffding, W. (1973) On the centering of a simple linear rank statistic. Ann. Statist. 1, 54–66.

    Article  MathSciNet  MATH  Google Scholar 

  35. Karlin, S. (1974) Inequalities For symmetric sampling plans I. Ann. Statist. 2 1005–1094.

    Article  MathSciNet  Google Scholar 

  36. Korolouk, V.S. and Borovskikh, Yu. V. (1989). Theory of U-Statistics (in Russian). Kiev.

    Google Scholar 

  37. LeCam, L. (1960). Locally asymptotically normal families of distributions. Univ. Calif. Publ. Statist. 3, 37–98.

    MathSciNet  Google Scholar 

  38. Lee, A.J. (1991). U-Statistics: Theory and Practice. Marcel Dekker, New York.

    Google Scholar 

  39. Lehmann, E.L. (1953). The power of rank tests. Ann. Math. Statist. 24, 23–43.

    Article  MATH  Google Scholar 

  40. Lehmann, E.L. (1966). Some concepts of dependence. Ann. Math. Statist. 37, 1137–1153.

    Article  MathSciNet  MATH  Google Scholar 

  41. Motoo, M. (1957). On the Hoeffding’s combinatorial central limit theorem. Ann. Inst. Statist. Math. 8, 145–154.

    Article  MathSciNet  MATH  Google Scholar 

  42. Noether, G.E. (1949). On a theorem by Wald and Wolfowitz. Ann. Math. Statist. 20, 455–458.

    Article  MathSciNet  MATH  Google Scholar 

  43. Noether, G.E. (1955). On a theorem of Pitman. Ann. Math. Statist. 26, 64–68.

    Article  MathSciNet  MATH  Google Scholar 

  44. Pitman, E.J.G. (1948). Notes on Nonparametric Statistical Inference (mimeographed), Columbia Univ. New York.

    Google Scholar 

  45. Pledger, G. and Proschan, F. (1971). Comparisons of order statistics from heterogeneous distributions. In Optimization Methods in Statistics (ed: J.S. Rustagi) Academic Press, New York, pp. 89–113.

    Google Scholar 

  46. Puri, M.L. and Sen, P.K. (1971). Nonparametric Methods in Multivariate Analysis. John Wiley, New York.

    Google Scholar 

  47. Puri, M.L. and Sen, P.K. (1985). Nonparametric Methods in General Linear Models. John Wiley, New York.

    Google Scholar 

  48. Pyke, R. and Shorack, G.R. (1968). Weak convergence of a two-sample empirical process and an approach to Chernoff-Savage theorems. Ann. Math. Statist. 39, 755–771.

    Article  MathSciNet  MATH  Google Scholar 

  49. Quade, D. (1965). On the asymptotic power of the one-sample Kolmogorov-Smirnov test. Ann. Math. Statist. 36, 1000–1018.

    Article  MathSciNet  MATH  Google Scholar 

  50. Rinott, Y. (1973). Multivariate majorization and rearrangement inequalities with some applications to probability and statistics. Israel J. Math. 15, 60–77.

    Article  MathSciNet  MATH  Google Scholar 

  51. Rosen, B. (1967). On an inequality of Hoeffding. Ann. Math. Statist. 38, 382–392.

    Article  MathSciNet  MATH  Google Scholar 

  52. Samuels, S.M. (1965). On the number of successes in independent trials. Ann. Math. Statist. 36, 1272–1278.

    Article  MathSciNet  MATH  Google Scholar 

  53. Schéffe, H. (1943). Statistical inference in the nonparametric case. Ann. Math. Statist. 14, 305–332.

    Article  MATH  Google Scholar 

  54. Sen, P.K. (1960). On some convergence properties of U-statistics. Calcutta Statist Assoc. Bull. 10, 1–18.

    MathSciNet  MATH  Google Scholar 

  55. Sen, P.K. (1970). A note on order statistics from heterogeneous distributions. Ann. Math. Statist. 41, 2137–2139.

    Article  MathSciNet  MATH  Google Scholar 

  56. Sen, P.K. (1981). Sequential Nonparametrics: Invariance Principles and Statistical Inference. John Wiley, New York.

    Google Scholar 

  57. Sen, P.K. (1983). On permutational central limit theorems for general multivariate linear rank statistics. Sankhya, Ser. A, 45, 141–149.

    MathSciNet  MATH  Google Scholar 

  58. Sen, P.K. (1992). Editorial introduction to “A class of statistics with asymptotically normal distribution, by W. Hoeffding, Annals of Mathematical Statistics, 19 (1948), 293-325,” in Breakthrough in Statistics 1890-1989 Volume 1, edited by S. Kotz and N.L. Johnson, Springer-Verlag, New York, pp. 299–307.

    Chapter  Google Scholar 

  59. Terry, M.E. (1952). Some rank order tests which are most powerful against specific parametric alternatives. Ann. Math. Statist. 23, 346–366.

    Article  MathSciNet  MATH  Google Scholar 

  60. van Zwet, W.R. (1984). A Berry-Esseen bound for symmetric statistics. Z. Wahr. verw. Geb. 66, 425–440.

    Article  MATH  Google Scholar 

  61. von Mises, R. (1947). On the asymptotic distribution of differentiable statistical functions. Ann. Math. Statist. 18, 309–348.

    Article  MATH  Google Scholar 

  62. Wald, A. and Wolfowitz, J. (1944). Statistical tests based on the permutation of the observations. Ann. Math. Statist. 15, 358–372.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of North Carolina at Chapel Hill, USA

    Pranab K. Sen

Authors
  1. Pranab K. Sen
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Division of Mathematics and Statisics, CSIRO, E6B, Macquarie University Campus, North Ryde, NSW, 2113, Australia

    N. I. Fisher

  2. Department of Statistics, University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599, USA

    P. K. Sen

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer Science+Business Media New York

About this chapter

Cite this chapter

Sen, P.K. (1994). The Impact of Wassily Hoeffding’s Research on Nonparametrics. In: Fisher, N.I., Sen, P.K. (eds) The Collected Works of Wassily Hoeffding. Springer Series in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0865-5_3

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-0865-5_3

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-6926-7

  • Online ISBN: 978-1-4612-0865-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation