# Hoeffding decomposition

Let $X _ {1} \dots X _ {N}$ be independent identically distributed random functions with values in a measurable space $( E, {\mathcal E} )$( cf. Random variable). For $m < N$, let

$$h : {E ^ {m} } \rightarrow \mathbf R$$

be a measurable symmetric function in $m$ variables and consider the $U$- statistics (cf. $U$- statistic)

$$U _ {N} ( h ) = { \frac{1}{\left ( \begin{array}{c} N \\ m \end{array} \right ) } } \sum _ {1 \leq i _ {1} < \dots < i _ {m} \leq N } h ( X _ {i _ {1} } \dots X _ {i _ {m} } ) .$$

The following theorem is called Hoeffding's decomposition theorem, and the representation of the $U$- statistic as in the theorem is called the Hoeffding decomposition of $U _ {N} ( h )$( see [a1]):

$$U _ {N} ( h ) = \sum _ {c = 0 } ^ { m } \left ( \begin{array}{c} m \\ c \end{array} \right ) U _ {N} ( h _ {c} ) ,$$

where ${h _ {c} } : {E ^ {c} } \rightarrow \mathbf R$ is a symmetric function in $c$ arguments and where the $U$- statistics $U _ {N} ( h _ {c} )$ are degenerate, pairwise orthogonal in $L _ {2}$( uncorrelated) and satisfy

$${\mathsf E} ( U _ {N} ( h _ {c} ) ) ^ {2} = {\mathsf E} ( h _ {c} ( X _ {1} \dots X _ {c} ) ) ^ {2} .$$

The symmetric functions $h _ {c}$ are defined as follows:

$$h _ {c} ( x _ {1} \dots x _ {c} ) = \sum _ {k = 0 } ^ { c } ( - 1 ) ^ {c - k } \times$$

$$\times \sum _ {1 \leq l _ {1} < \dots < l _ {k} \leq c } E ( h ( x _ {l _ {1} } \dots x _ {l _ {k} } ,X _ {1} \dots X _ {m - k } ) ) .$$

Extensions of this decomposition are known for the multi-sample case [a4], under various "uncomplete" summation procedures in the definition of a $U$- statistic, in some dependent situations and for non-identical distributions [a3]. There are also versions of the theorem for symmetric functions that have values in a Banach space.

The decomposition theorem permits one to easily calculate the variance of $U$- statistics. Since $U _ {N} ( h _ {0} ) = {\mathsf E} h ( X _ {1} \dots X _ {m} )$ and since $U _ {N} ( h _ {1} )$ is a sum of centred independent identically distributed random variables, the central limit theorem for non-degenerate $U$- statistics is an immediate consequence of the Hoeffding decomposition (cf. also Central limit theorem).

The terminology goes back to [a2].

#### References

 [a1] M. Denker, "Asymptotic distribution theory in nonparametric statistics" , Advanced Lectures in Mathematics , F. Vieweg (1985) [a2] W. Hoeffding, "A class of statistics with asymptotically normal distribution" Ann. Math. Stat. , 19 (1948) pp. 293–325 [a3] A.J. Lee, "U-statistics. Theory and practice" , Statistics textbooks and monographs , 110 , M. Dekker (1990) [a4] E.L. Lehmann, "Consistency and unbiasedness of certain nonparametric tests" Ann. Math. Stat. , 22 (1951) pp. 165–179
How to Cite This Entry:
Hoeffding decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hoeffding_decomposition&oldid=47242
This article was adapted from an original article by M. Denker (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article