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Uniform in Bandwidth Consistency of Conditional U-statistics Adaptive to Intrinsic Dimension in Presence of Censored Data

Author

Listed:
  • Salim Bouzebda

    (Université de Technologie de Compiègne, LMAC (Laboratory of Applied Mathematics of Compiègne))

  • Thouria El-hadjali

    (Université Frères Mentouri)

  • Anouar Abdeldjaoued Ferfache

    (Université de Technologie de Compiègne, LMAC (Laboratory of Applied Mathematics of Compiègne))

Abstract

U-statistics represent a fundamental class of statistics from modelling quantities of interest defined by multi-subject responses. U-statistics generalize the empirical mean of a random variable X to sums over every m-tuple of distinct observations of Stute (Ann. Probab. 19, 812–825 1991) introduced a class of so-called conditional U-statistics, which may be viewed as a generalization of the Nadaraya-Watson estimates of a regression function. Stute proved their strong pointwise consistency to: r ( t ) : = E [ φ ( Y 1 , … , Y m ) | ( X 1 , … , X m ) = t ] , for t ∈ ℝ d m . $$r(\mathbf{ t}):=\mathbb{E}[\varphi(Y_{1},\ldots,Y_{m})|(X_{1},\ldots,X_{m})=\mathbf{t}], ~~\text{for}~~\mathbf{ t}\in \mathbb{R}^{dm}.$$ We apply the methods developed in Dony and Mason (Bernoulli 14(4), 1108–1133 2008) to establish uniform in t and in bandwidth consistency (i.e., h, h ∈ [an,bn] where 0

Suggested Citation

  • Salim Bouzebda & Thouria El-hadjali & Anouar Abdeldjaoued Ferfache, 2023. "Uniform in Bandwidth Consistency of Conditional U-statistics Adaptive to Intrinsic Dimension in Presence of Censored Data," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(2), pages 1548-1606, August.
    • Handle: RePEc:spr:sankha:v:85:y:2023:i:2:d:10.1007_s13171-022-00301-7
    DOI: 10.1007/s13171-022-00301-7
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