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Asymptotics of randomly weighted u- and v-statistics: Application to bootstrap

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  • Csörgő, Miklós
  • Nasari, Masoud M.

Abstract

This paper is mainly concerned with asymptotic studies of weighted bootstrap for u- and v-statistics. We derive the consistency of the weighted bootstrap u- and v-statistics, based on i.i.d. and non i.i.d. observations from some more general results which we first establish for sums of randomly weighted arrays of random variables. Some of the results in this paper significantly extend some well-known results on consistency of u-statistics and also consistency of sums of arrays of random variables. We also employ a new approach to conditioning to derive a conditional central limit theorem (CLT) for weighted bootstrap u- and v-statistics, assuming the same conditions as the classical CLT for regular u- and v-statistics.

Suggested Citation

  • Csörgő, Miklós & Nasari, Masoud M., 2013. "Asymptotics of randomly weighted u- and v-statistics: Application to bootstrap," Journal of Multivariate Analysis, Elsevier, vol. 121(C), pages 176-192.
    • Handle: RePEc:eee:jmvana:v:121:y:2013:i:c:p:176-192
    DOI: 10.1016/j.jmva.2013.07.008
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    References listed on IDEAS

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    1. Burke, Murray D. & Gombay, Edit, 1988. "On goodness-of-fit and the bootstrap," Statistics & Probability Letters, Elsevier, vol. 6(5), pages 287-293, April.
    2. Arenal-Gutiérrez, Eusebio & Matrán, Carlos & Cuesta-Albertos, Juan A., 1996. "On the unconditional strong law of large numbers for the bootstrap mean," Statistics & Probability Letters, Elsevier, vol. 27(1), pages 49-60, March.
    3. Wang, Qiying & Jing, Bing-Yi, 2004. "Weighted bootstrap for U-statistics," Journal of Multivariate Analysis, Elsevier, vol. 91(2), pages 177-198, November.
    4. Rosalsky, Andrew & Sreehari, M., 1998. "On the limiting behavior of randomly weighted partial sums," Statistics & Probability Letters, Elsevier, vol. 40(4), pages 403-410, November.
    5. Athreya, K. B., 1983. "Strong law for the bootstrap," Statistics & Probability Letters, Elsevier, vol. 1(3), pages 147-150, March.
    6. Nasari, Masoud M., 2012. "Strong law of large numbers for weighted U-statistics: Application to incomplete U-statistics," Statistics & Probability Letters, Elsevier, vol. 82(6), pages 1208-1217.
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    Cited by:

    1. Gajek, Lesław & Krajewska, Elżbieta, 2020. "Approximating sums of products of dependent random variables," Statistics & Probability Letters, Elsevier, vol. 164(C).

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