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Asymptotic normality of variance estimator in a heteroscedastic model with dependent errors

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  • Han-Ying Liang
  • Ya-Mei Liu

Abstract

Consider the heteroscedastic regression model Yni=g(xni)+σniεni (1≤i≤n), where , the design points (xni, uni) are known and nonrandom, g(·) and f(·) are unknown functions defined on [0, 1], and the random errors {εni, 1≤i≤n} are assumed to have the same distribution as {ξi, 1≤i≤n}, which is a stationary and α-mixing time series with Eξi=0. Under appropriate conditions, we study the asymptotic normality of an estimator of the function f(·). At the same time, we derive a Berry–Esseen-type bound for the estimator. As a corollary, by making a certain choice of the weights, the Berry–Esseen-type bound of the estimator can attain O(n−1/12(log n)−1/3). Finite sample behaviour of this estimator is investigated too.

Suggested Citation

  • Han-Ying Liang & Ya-Mei Liu, 2011. "Asymptotic normality of variance estimator in a heteroscedastic model with dependent errors," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(2), pages 351-365.
    • Handle: RePEc:taf:gnstxx:v:23:y:2011:i:2:p:351-365
    DOI: 10.1080/10485252.2011.552721
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    References listed on IDEAS

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    1. Liang, Han-Ying & Fan, Guo-Liang, 2009. "Berry-Esseen type bounds of estimators in a semiparametric model with linear process errors," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 1-15, January.
    2. Roussas, George G. & Tran, Lanh T. & Ioannides, D. A., 1992. "Fixed design regression for time series: Asymptotic normality," Journal of Multivariate Analysis, Elsevier, vol. 40(2), pages 262-291, February.
    3. Georgiev, Alexander A., 1988. "Consistent nonparametric multiple regression: The fixed design case," Journal of Multivariate Analysis, Elsevier, vol. 25(1), pages 100-110, April.
    4. Liang, Han-Ying & Jing, Bing-Yi, 2005. "Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences," Journal of Multivariate Analysis, Elsevier, vol. 95(2), pages 227-245, August.
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