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