Introduction to Hoeffding (1948) A Class of Statistics with Asymptotically Normal Distribution
Part of the
Springer Series in Statistics
book series (SSS)
Wassily Hoeffding was born on June 12, 1914 in Mustamaki, Finland, near St. Petersburg (now Leningrad), USSR. His parents were of Danish origin; his father was an economist and his mother had studied medicine. Although at that time, Finland was a part of the Russian Empire, the Bolshevik movement was quite intense and consolidated under the Lenin dictatorship in the Civil War of 1918-20. The Hoeffding family left Russia for Denmark in 1920, and four years later, they moved on to Berlin. In 1933, Wassily finished high school and went on to college to study economics. However, a year later, he gave up economics and entered Berlin University to study mathematics. He earned a Ph.D. degree from Berlin University in 1940 with a dissertation in correlation theory, which dealt with some properties of bivariate distributions that are invariant under arbitrary monotone transformations of the margins. In this context, he studied some (mostly, descriptive) aspects of rank correlations, and a few years later, while investigating the sampling aspects of such measures, he formulated in a remarkably general and highly original form the general distribution theory of symmetric, unbiased estimators (which he termed U-statistics). This is depicted in this outstanding article (under commentary). During World War II, Wassily worked in Berlin partly as an editorial assistant and partly as a research assistant in actuarial science. In September 1946, he was able to immigrate to the United States, and there he started attending lectures at Columbia University in New York. In 1947, he was invited by Harold Hotelling to join the newly established department of Statistics at the University of North Carolina at Chapel Hill. Since then Wassily had been in Chapel Hill with occasional visits to other campuses (Columbia University and Cornell University in New York, the Steklov Institute in Leningrad, USSR, and the Indian Statistical Institute in Calcutta, among others). In 1979, he retired from active service. He died on February 28, 1991 at Chapel Hill.
KeywordsCovariance Germinate Estima Banner
Berk, R.H. (1956). Limiting behavior of posterior distributions when the model is incorrect, Ann. Math. Statist.
, 51–58.MathSciNetCrossRefGoogle Scholar
Dwass, M. (1955). On the asymptotic normality of some statistics used in nonparametric tests, Ann. Math. Statist.
, 334–339.MathSciNetMATHCrossRefGoogle Scholar
Gani, J. (ed.) (1982). The Making of Statisticians
. Springer-Verlag, New York.MATHGoogle Scholar
Gregory, G.G. (1977). Large sample theory for U-statistics and tests of fit, Ann. Statist.
, 110–123.MathSciNetMATHCrossRefGoogle Scholar
Hájek, J. (1968). Asymptotic normality of simple linear rank statistics under alternatives, Ann. Math. Statist.
, 325–346.MathSciNetMATHCrossRefGoogle Scholar
Halmos, P.R. (1946). The theory of unbiased estimation, Ann. Math. Statist.
, 34–43.MathSciNetMATHCrossRefGoogle Scholar
Hoeffding, W. (1948). A nonparametric test for independence, Ann. Math. Statist.
, 546–557.MathSciNetMATHCrossRefGoogle Scholar
Hoeffding, W. (1961). The strong law of large numbers for U-statistics. Institute of Statistics University of North Carolina Mimeo Series No. 302.Google Scholar
Keenan, D.M. (1983). Limiting behavior of functionals of the sample spectral distribution, Ann. Statist.
, 1206–1217.MathSciNetMATHGoogle Scholar
Kingman, J.F.C. (1969). An Ergodic theorem, Bull. Lon. Math. Soc.
, 339–340.MathSciNetMATHCrossRefGoogle Scholar
Koroljuk, Yu.S., and Borovskih, Yu.V. (1989). Teorija U-Statistik
(in Russian). Naukova Dumka, Kiev.Google Scholar
Lehmann, E.L. (1951). Consistency and unbiasedness of certain nonparametric tests, Ann. Math. Statist.
, 165–179.MATHCrossRefGoogle Scholar
Miller, R.G., Jr., and Sen, P.K. (1972). Weak convergence of U-statistics and von Mises’ differentiable statistical functions, Ann. Math. Statist.
, 31–41.MathSciNetMATHCrossRefGoogle Scholar
Puri, M.L. and Sen, P.K. (1971). Nonparametric Methods in Multivariate Analysis
. Wiley, New York.MATHGoogle Scholar
Sen, P.K. (1981). Sequential Nonparametrics: Invariance Principles and Statistical Inference
. Wiley, New York.MATHGoogle Scholar
Sen, P.K. (1988). Functional jackknifing: Rationality and general asymptotics, Ann. Statist.
, 450–469.MathSciNetMATHCrossRefGoogle Scholar
Serfling, R.J. (1980). Approximation Theorems of Mathematical Statistics
. Wiley, New York.MATHCrossRefGoogle Scholar
Van Zwet, W.R. (1984). A Berry-Esseen bound for symmetric statistics, Zeit. Wahrsch. verw. Gebiete
, 425–440.MATHCrossRefGoogle Scholar
von Mises, R.V. (1947). On the asymptotic distribution of differentiable statistical functions, Ann. Math. Statist.
, 309–348.MATHCrossRefGoogle Scholar
© Springer Science+Business Media New York 1992