Asymptotic Expansions for Symmetric Statistics with Degenerate Kernels

Shuya Kanagawa ()
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Shuya Kanagawa: Department of Mathematics, Tokyo Gakugei University, 4-1-1 Nukuikita-machi, Tokyo 184-8501, Japan

Mathematics, 2022, vol. 10, issue 21, 1-10

Abstract: Asymptotic expansions for U-statistics and V-statistics with degenerate kernels are investigated, respectively, and the remainder term O ( n 1 − p / 2 ) , for some p ≥ 4 , is shown in both cases. From the results, it is obtained that asymptotic expansions for the Cram e ´ r–von Mises statistics of the uniform distribution U ( 0 , 1 ) hold with the remainder term O n 1 − p / 2 for any p ≥ 4 . The scheme of the proof is based on three steps. The first one is the almost sure convergence in a Fourier series expansion of the kernel function u ( x , y ) . The key condition for the convergence is the nuclearity of a linear operator T u defined by the kernel function. The second one is a representation of U-statistics or V-statistics by single sums of Hilbert space valued random variables. The third one is to apply asymptotic expansions for single sums of Hilbert space valued random variables.

Keywords: U-statistics; V-statistics; asymptotic expansion; integral kernel; nuclearity (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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