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Local Walsh-average regression

Author

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  • Feng, Long
  • Zou, Changliang
  • Wang, Zhaojun

Abstract

Local polynomial regression is widely used for nonparametric regression. However, the efficiency of least squares (LS) based methods is adversely affected by outlying observations and heavy tailed distributions. On the other hand, the least absolute deviation (LAD) estimator is more robust, but may be inefficient for many distributions of interest. Kai et al. (2010) [13] propose a nonparametric regression technique called local composite quantile regression (LCQR) smoothing to improve local polynomial regression further. However, the performance of LCQR depends on the choice of the number of quantiles to combine, a meta parameter which plays a vital role in balancing the performance of LS and LAD based methods. To overcome this issue, we propose a novel method termed the local Walsh-average regression (LWAR) estimator by minimizing a locally Walsh-average based loss function. Under the same assumptions in Kai et al. (2010) [13], we theoretically show that the proposed estimator is highly efficient across a wide spectrum of distributions. Its asymptotic relative efficiency with respect to the LS based method is closely related to that of the signed-rank Wilcoxon test in comparison with the t-test. Both of the theoretical and numerical results demonstrate that the performance of the new approach and LCQR is at least comparable in estimating the nonparametric regression function or its derivatives and in some cases the new approach performs better than the LCQR with commonly recommended number of quantiles, especially for estimating the regression function.

Suggested Citation

  • Feng, Long & Zou, Changliang & Wang, Zhaojun, 2012. "Local Walsh-average regression," Journal of Multivariate Analysis, Elsevier, vol. 106(C), pages 36-48.
  • Handle: RePEc:eee:jmvana:v:106:y:2012:i:c:p:36-48
    DOI: 10.1016/j.jmva.2011.12.003
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    References listed on IDEAS

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    1. Wang, Hansheng & Xia, Yingcun, 2009. "Shrinkage Estimation of the Varying Coefficient Model," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 747-757.
    2. Terpstra, Jeff T. & McKean, Joseph W., 2005. "Rank-Based Analysis of Linear Models Using R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 14(i07).
    3. Bo Kai & Runze Li & Hui Zou, 2010. "Local composite quantile regression smoothing: an efficient and safe alternative to local polynomial regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(1), pages 49-69, January.
    4. Wang, Lan & Kai, Bo & Li, Runze, 2009. "Local Rank Inference for Varying Coefficient Models," Journal of the American Statistical Association, American Statistical Association, vol. 104(488), pages 1631-1645.
    5. Pollard, David, 1991. "Asymptotics for Least Absolute Deviation Regression Estimators," Econometric Theory, Cambridge University Press, vol. 7(2), pages 186-199, June.
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    Citations

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    Cited by:

    1. Shang, Suoping & Zou, Changliang & Wang, Zhaojun, 2012. "Local Walsh-average regression for semiparametric varying-coefficient models," Statistics & Probability Letters, Elsevier, vol. 82(10), pages 1815-1822.
    2. Jing Sun & Lu Lin, 2014. "Local rank estimation and related test for varying-coefficient partially linear models," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 26(1), pages 187-206, March.
    3. Niu, Cuizhen & Guo, Xu & Li, Yong & Zhu, Lixing, 2018. "Pairwise distance-based tests for conditional symmetry," Computational Statistics & Data Analysis, Elsevier, vol. 128(C), pages 145-162.
    4. Long Feng & Changliang Zou & Zhaojun Wang & Lixing Zhu, 2015. "Robust comparison of regression curves," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(1), pages 185-204, March.
    5. Ayub, Kanwal & Song, Weixing & Shi, Jianhong, 2022. "Extrapolation estimation in parametric regression models with measurement error," Computational Statistics & Data Analysis, Elsevier, vol. 172(C).
    6. Yunquan Song & Hang Su & Minmin Zhan, 2024. "Local Walsh-average-based Estimation and Variable Selection for Spatial Single-index Autoregressive Models," Networks and Spatial Economics, Springer, vol. 24(2), pages 313-339, June.

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