Normal approximation for finite-populationU-statistics

Lincheng, Zhao; Chen, Xiru

doi:10.1007/bf02019152

Cited by 38 publications

(28 citation statements)

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“…M is different from that deduced in Zhao and Chen (1990) or Bloznelis and Götze (2001) for the case of A being a finite set with cardinality S > 2M , endowed with the counting measure α. However, Corollary 4 ensures that the results implied by Theorem 11 and those in the references above are equivalent.…”

Section: Hoeffding Decompositions For Gus (Statements)contrasting

confidence: 64%

“…However, it has been completely solved in only two cases: when X is a sequence of i.i.d. random variables [as first proved in Hoeffding (1948), see, e.g., Hajek (1968, Efron and Stein (1981), Karlin and Rinott (1982), Takemura (1983), Vitale (1990), Bentkus, Götze and van Zwet (1997) and the references therein], and when X is a collection of N − 1 extractions without replacement from a finite population [see Zhao and Chen (1990) and Götze (2001, 2002)], and in both instances, the degenerated U -statistics T i turn out to be linear combinations of well chosen conditional expectations of T .…”

Section: = (Xmentioning

confidence: 99%

“…. , a S } [this is the case studied in Zhao and Chen (1990) and Götze (2001, 2002)]; when c > 0 and N is infinite, X (α,c) is a generalized Pólya urn sequence whose directing measure [in the terminology of Aldous (1983)] is a Dirichlet-Ferguson process on (A, A) with parameter α(·)/c [the reader is referred to Ferguson (1973), Blackwell and MacQueen (1973), Blackwell (1973) and Ferguson (1974) for definitions, proofs of the above claims and discussions of the relevance of such objects in Bayesian nonparametric statistics; see also Pitman (1996) for a rich survey of some recent developments of Pólya urn processes]. Note also that, in all cases, the law of X (α,c) is characterized by the following two facts: (i) for every j < N , P(X (α,c) j…”

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confidence: 99%

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Hoeffding-ANOVA decompositions for symmetric statistics of exchangeable observations

Peccati¹

2004

Ann. Probab.

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Consider a (possibly infinite) exchangeable sequence X = {Xn : 1 ≤ n < N }, where N ∈ N ∪ {∞}, with values in a Borel space (A, A), and note Xn = (X1, . . . , Xn). We say that X is Hoeffding decomposable if, for each n, every square integrable, centered and symmetric statistic based on Xn can be written as an orthogonal sum of n Ustatistics with degenerated and symmetric kernels of increasing order. The only two examples of Hoeffding decomposable sequences studied in the literature are i.i.d. random variables and extractions without replacement from a finite population. In the first part of the paper we establish a necessary and sufficient condition for an exchangeable sequence to be Hoeffding decomposable, that is, called weak independence. We show that not every exchangeable sequence is weakly independent, and, therefore, that not every exchangeable sequence is Hoeffding decomposable. In the second part we apply our results to a class of exchangeable and weakly independent random vectors X (α,c) n = (X (α,c) 1 , . . . , X (α,c) n ) whose law is characterized by a positive and finite measure α(•) on A and by a real constant c. For instance, if c = 0, X (α,c) n

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Section: Hoeffding Decompositions For Gus (Statements)contrasting

confidence: 64%

Section: = (Xmentioning

confidence: 99%

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confidence: 99%

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Hoeffding-ANOVA decompositions for symmetric statistics of exchangeable observations

Peccati¹

2004

Ann. Probab.

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“…Using Hoeffding's decomposition Bentkus, Götze and van Zwet [4] constructed second order approximations to distribution functions of general asymptotically linear symmetric statistics of independent observations. For samples drawn without replacement Hoeffding's decomposition was studied in Bloznelis and Götze [9] and Zhao and Chen [23]. The one-term Edgeworth expansion for finite population Student statistic is shown in Babu and Singh [3].…”

Section: Preliminariesmentioning

confidence: 99%

Bootstrap Approximation to Distributions of Finite Population U-statistics

Bloznelis

2007

Acta Appl Math

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We show that the without replacement bootstrap of Booth, Butler and Hall (J. Am. Stat. Assoc. 89, 1282-1289, 1994 provides second order correct approximation to the distribution function of a Studentized U-statistic based on simple random sample drawn without replacement. In order to achieve similar approximation accuracy for the bootstrap procedure due to Bickel and Freedman (Ann. Stat. 12, 470-482, 1984) and Chao and Lo (Sankhya Ser. A 47, 399-405, 1985) we introduce randomized adjustments to the resampling fraction.

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“…See El-Dakkak and Peccati [7] and Peccati [9] for some general statements; see Bloznelis [2], Bloznelis and Götze [3,4] and Zhao and Chen [14] for a comprehensive analysis of Hoeffding decompositions associated with extractions without replacement from a finite population.…”

Section: Introductionmentioning

confidence: 99%