Limit distributions of conditional U-statistics

Prakasa Rao, B. L. S. ; Sen, Arusharka (1995) Limit distributions of conditional U-statistics Journal of Theoretical Probability, 8 (2). pp. 261-301. ISSN 0894-9840

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Let{(Xn, Yn)}n ≥ 1 be a sequence of i.i.d. bi-variate vectors. In this article, we study the possible limit distributions of Uh n (t), the so-called conditional U-statistics, introduced by Stute. (10) They are estimators of functions of the form mh(t)=E{h(Y 1,...,Yk)|X1=t l,...,Xk = tk},t=(t 1,...,tk) ∈Rk where E|h|<∞. Here t is fixed. In case tl=...=tk=t (say), we describe the limiting random variables as multiple Wiener integrals with respect to Pt, the conditional distribution of Y, given X=t. When ti, 1≤i≤k, are not all equal, we introduce and use a slightly generalized version of a multiple Wiener integral.

Item Type:Article
Source:Coptright of this article belongs to Springer.
Keywords:Nonparametric Regression Estimation; Conditional U-statistics; Limit Distribution; Symmetric Tensor Product; Multiple Wiener Integral; Dynkin-Mandelbaum Decomposition; Scheffe's Lemma
ID Code:37207
Deposited On:26 Apr 2011 10:48
Last Modified:26 Apr 2011 10:48

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