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A class of asymptotically normal degenerate quasi U-statistics

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Abstract

Some quasi U-statistics, unlike other variants of U-statistics, arising in distance based tests for homogeneity of groups, have first-order stationary kernels of degree 2, and yet they enjoy asymptotic normality under suitable hypotheses of invariance. Central limit theorems for a more general class of quasi U-statistics with possibly higher order stationarity (and degree) are formulated with the aid of appropriate martingale (array) characterizations as well as permutational invariance structures.

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Authors and Affiliations

  1. Departamento de Estatística, Universidade Estadual de Campinas, Cidade Universitária Zeferino Vaz, Cx. Postal 6065, Campinas, SP, CEP 13083-970, Brazil

    Aluísio Pinheiro & Hildete P. Pinheiro

  2. Department of Biostatistics and Statistics and Operations Research, School of Public Health, UNC at Chapel Hill, 3105E McGavran-Greenberg, CB# 7420, Chapel Hill, NC, 27599-7420, USA

    Pranab Kumar Sen

Authors
  1. Aluísio Pinheiro
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  2. Pranab Kumar Sen
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  3. Hildete P. Pinheiro
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Correspondence to Aluísio Pinheiro.

Additional information

Acknowledgment of support: This research was funded in part by FAPESP (08/51097-6; 08/09286-6;09/14176-8) and CNPq (306993/2008-2; 480919/2009-7; 306240/2009-2).

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Pinheiro, A., Sen, P.K. & Pinheiro, H.P. A class of asymptotically normal degenerate quasi U-statistics. Ann Inst Stat Math 63, 1165–1182 (2011). https://doi.org/10.1007/s10463-010-0271-z

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  • DOI: https://doi.org/10.1007/s10463-010-0271-z

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