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The central limit theorem for degenerate variable -statistics under dependence

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  • Tae Kim
  • Zhi-Ming Luo
  • Chiho Kim

Abstract

The central limit theorem (CLT) for degenerate U-statistics with a variable symmetric kernel function has been studied under dependence by many authors, since it has many important applications in nonparametric estimation and testing problems [see, e.g. Takahata, H., and Yoshihara, K. (1987), ‘Central Limit Theorems for Integrated Square Error of Nonparametric Density Estimators Based on a Absolutely Regular Random Sequences’, Yokohama Mathematical Journal, 35, 95–111; Yoshihara, K. (1989), ‘Limiting Behavior of Generalized Quadratic Forms Generated by Absolutely Regular Sequences II’, Yokohama Mathematical Journal, 37, 109–123. Yoshihara, K. (1992), ‘Limiting Behavior of Generalized Quadratic Forms Generated by Absolutely Regular Sequences III’, Yokohama Mathematical Journal, 40, 1–9; Fan, J., and Li, Q. (1999), ‘Central Limit Theorem for Degenerate U-Statistics of Absolutely Regular Processes with Applications to Model Specification Testing’, Journal of Nonparametric Statistics, 10, 245–271; Gao, J., and King, M.L. (2004), ‘Adaptive Testing in Continuous-time Diffusion Models’, Econometric Theory, 20, 844–882; Gao, J. (2007), Nonlinear Time Series: Semiparametric and Nonparametric Methods, Chapman & Hall/CRC; Gao, J., and Hong, Y. (2008), ‘Central Limit Theorem for Generalized U-statistics with Applications in Nonparametric Specification’, Journal of Nonparametric Statistics, 20, 61–76]. In this paper, we provide an improved version with the asymmetric kernel method which is quite useful for application to nonparametric methods in various situations. As an illustration of the usefulness of our result, CLTs for quadratic errors of a nonparametric density estimator are developed under dependency, which is meaningful in its own right.

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  • Tae Kim & Zhi-Ming Luo & Chiho Kim, 2011. "The central limit theorem for degenerate variable -statistics under dependence," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(3), pages 683-699.
    • Handle: RePEc:taf:gnstxx:v:23:y:2011:i:3:p:683-699
    DOI: 10.1080/10485252.2011.556193
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    1. Fan, Yanqin & Li, Qi, 1996. "Consistent Model Specification Tests: Omitted Variables and Semiparametric Functional Forms," Econometrica, Econometric Society, vol. 64(4), pages 865-890, July.
    2. Kim, T. Y. & Cox, D. D., 1995. "Asymptotic Behaviors of Some Measures of Accuracy in Nonparametric Curve Estimation with Dependent Observations," Journal of Multivariate Analysis, Elsevier, vol. 53(1), pages 67-93, April.
    3. Hall, Peter, 1984. "Central limit theorem for integrated square error of multivariate nonparametric density estimators," Journal of Multivariate Analysis, Elsevier, vol. 14(1), pages 1-16, February.
    4. Gao, Jiti, 2007. "Nonlinear time series: semiparametric and nonparametric methods," MPRA Paper 39563, University Library of Munich, Germany, revised 01 Sep 2007.
    5. Jiti Gao & Yongmiao Hong, 2008. "Central limit theorems for generalized -statistics with applications in nonparametric specification," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(1), pages 61-76.
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    Cited by:

    1. Rauf Ahmad, M. & Pavlenko, Tatjana, 2018. "A U-classifier for high-dimensional data under non-normality," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 269-283.
    2. David Källberg & Nikolaj Leonenko & Oleg Seleznjev, 2014. "Statistical estimation of quadratic Rényi entropy for a stationary m -dependent sequence," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 26(2), pages 385-411, June.
    3. Fu, Zhonghao & Hong, Yongmiao & Su, Liangjun & Wang, Xia, 2023. "Specification tests for time-varying coefficient models," Journal of Econometrics, Elsevier, vol. 235(2), pages 720-744.

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