The class of U-statistics was introduced in the seminal paper of Hoeffding (1948), motivated by the work of Halmos (1946) on unbiased estimators. Let X 1, X 2, …, X n be independent and identically distributed observations from a distribution F(x, θ), where θ is some parameter of interest. If g is a function of m variables such that Eg(X 1, …, X m ) = θ then a natural unbiased estimator for θ can be constructed by evaluating g at all subsets of size m that can be formed from the observations and then averaging these values. Write
where n (m) = n(n − 1)…(n − m + 1) and the sum is over all m-tuples (i 1, …, i m ) of distinct elements of {1, 2, …, n}. Note we can consider the symmetric function formed by first averaging over all permutations of a given set of indices, so \(h({x}_{1},\ldots ,{x}_{m}) = {(m!)}^{-1} \sum \nolimits g({x}_{{i}_{1}},\ldots ,{x}_{{i}_{m}}),\) where the sum is over all m!...
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References and Further Reading
de la Peña VH, Giné E (1999) Decoupling. From dependence to independence. Springer, New York
Dynkin EB, Mandelbaum A (1983) Symmetric statistics, Poisson point processes and multiple Weiner integrals. Ann Stat 11:739–745
Halmos PR (1946) The theory of unbiased estimators. Ann Math Stat 17:34–43
Hoeffding W (1948) A class of statistics with asymptotically normal distribution. Ann Math Stat 19:293–325
Koroljuk VS, Borovskikh YV (1994) Theory of U-statistics. Kluwer, Dordrecht
Lee AJ (1990) U-statistics. Theory and practice. Marcel Dekker, New York
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Weber, N.C. (2011). U-Statistics. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_607
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