IDEAS home Printed from https://ideas.repec.org/a/bla/istatr/v88y2020i1p142-154.html
   My bibliography  Save this article

Asymptotics of the Non‐parametric Function for B‐splines‐based Estimation in Partially Linear Models

Author

Listed:
  • Heng Lian

Abstract

We consider least squares method for partially linear models based on polynomial splines. We derive the asymptotic property for the estimator, focusing on the estimation of the non‐parametric function, in particular whether and how the estimation of the linear part will affect the non‐parametric part (the converse relation, that is, how the linear part will be affected by the non‐parametric part is much better known, which we will also review). One important goal along the way is to clarify the role of projection in semiparametric models, which was nevertheless a classical trick for proving the asymptotic normality of the linear part. A crucial question we try to answer is whether projection plays any role in the estimation of the non‐parametric function. The answer is both positive and negative depending on the direction along which to assess asymptotic normality. The style of writing of the paper is somewhat expository, and it also contains several new results not found in the current literature. Finally, we demonstrate in our numerical studies that construction of the pointwise confidence intervals for the non‐parametric function motivated by our theory improves upon those constructed by pretending the linear part is known.

Suggested Citation

  • Heng Lian, 2020. "Asymptotics of the Non‐parametric Function for B‐splines‐based Estimation in Partially Linear Models," International Statistical Review, International Statistical Institute, vol. 88(1), pages 142-154, April.
    • Handle: RePEc:bla:istatr:v:88:y:2020:i:1:p:142-154
    DOI: 10.1111/insr.12346
    as

    Download full text from publisher

    File URL: https://doi.org/10.1111/insr.12346
    Download Restriction: no
    File URL: https://libkey.io/10.1111/insr.12346?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Heng Lian & Xin Chen & Jian-Yi Yang, 2012. "Identification of Partially Linear Structure in Additive Models with an Application to Gene Expression Prediction from Sequences," Biometrics, The International Biometric Society, vol. 68(2), pages 437-445, June.
    2. Hardle, Wolfgang & LIang, Hua & Gao, Jiti, 2000. "Partially linear models," MPRA Paper 39562, University Library of Munich, Germany, revised 01 Sep 2000.
    3. Lian, Heng & Liang, Hua, 2013. "Generalized Additive Partial Linear Models With High-Dimensional Covariates," Econometric Theory, Cambridge University Press, vol. 29(6), pages 1136-1161, December.
    4. Lian, Heng & Liang, Hua, 2016. "Separation of linear and index covariates in partially linear single-index models," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 56-70.
    5. Li, Qi, 2000. "Efficient Estimation of Additive Partially Linear Models," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 41(4), pages 1073-1092, November.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Zhang, Jun & Lin, Bingqing & Zhou, Yan, 2021. "Kernel density estimation for partial linear multivariate responses models," Journal of Multivariate Analysis, Elsevier, vol. 185(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Lan, Wei & Ding, Yue & Fang, Zheng & Fang, Kuangnan, 2016. "Testing covariates in high dimension linear regression with latent factors," Journal of Multivariate Analysis, Elsevier, vol. 144(C), pages 25-37.
    2. Kim, Kun Ho & Chao, Shih-Kang & Härdle, Wolfgang Karl, 2020. "Simultaneous Inference of the Partially Linear Model with a Multivariate Unknown Function," IRTG 1792 Discussion Papers 2020-008, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    3. Dette, Holger & Marchlewski, Mareen, 2007. "A test for the parametric form of the variance function in apartial linear regression model," Technical Reports 2007,26, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    4. Hilafu, Haileab & Wu, Wenbo, 2017. "Partial projective resampling method for dimension reduction: With applications to partially linear models," Computational Statistics & Data Analysis, Elsevier, vol. 109(C), pages 1-14.
    5. Ferraccioli, Federico & Sangalli, Laura M. & Finos, Livio, 2022. "Some first inferential tools for spatial regression with differential regularization," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    6. Cui, Xia & Lu, Ying & Peng, Heng, 2017. "Estimation of partially linear regression models under the partial consistency property," Computational Statistics & Data Analysis, Elsevier, vol. 115(C), pages 103-121.
    7. Yuejin Zhou & Yebin Cheng & Wenlin Dai & Tiejun Tong, 2018. "Optimal difference-based estimation for partially linear models," Computational Statistics, Springer, vol. 33(2), pages 863-885, June.
    8. Zhu, Xuehu & Wang, Tao & Zhao, Junlong & Zhu, Lixing, 2017. "Inference for biased transformation models," Computational Statistics & Data Analysis, Elsevier, vol. 109(C), pages 105-120.
    9. Yayi Yan & Jiti Gao & Bin Peng, 2020. "A Class of Time-Varying Vector Moving Average Models: Nonparametric Kernel Estimation and Application," Papers 2010.01492, arXiv.org.
    10. Kim, Namhyun & W. Saart, Patrick, 2021. "Estimation in partially linear semiparametric models with parametric and/or nonparametric endogeneity," Cardiff Economics Working Papers E2021/9, Cardiff University, Cardiff Business School, Economics Section.
    11. Chaohua Dong & Jiti Gao & Bin Peng, 2015. "Partially Linear Panel Data Models with Cross-Sectional Dependence and Nonstationarity," Monash Econometrics and Business Statistics Working Papers 7/15, Monash University, Department of Econometrics and Business Statistics.
    12. Lu Lin & Lili Liu & Xia Cui & Kangning Wang, 2021. "A generalized semiparametric regression and its efficient estimation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(1), pages 1-24, March.
    13. Wang, Zhaoliang & Xue, Liugen & Liu, Juanfang, 2019. "Checking nonparametric component for partially nonlinear model with missing response," Statistics & Probability Letters, Elsevier, vol. 149(C), pages 1-8.
    14. Cai, Xizhen & Zhu, Yeying & Huang, Yuan & Ghosh, Debashis, 2022. "High-dimensional causal mediation analysis based on partial linear structural equation models," Computational Statistics & Data Analysis, Elsevier, vol. 174(C).
    15. Aifen Feng & Xiaogai Chang & Jingya Fan & Zhengfen Jin, 2023. "Application of LADMM and As-LADMM for a High-Dimensional Partially Linear Model," Mathematics, MDPI, vol. 11(19), pages 1-14, October.
    16. Nengxiang Ling & Germán Aneiros & Philippe Vieu, 2020. "kNN estimation in functional partial linear modeling," Statistical Papers, Springer, vol. 61(1), pages 423-444, February.
    17. Zhang, Yaowu & Zhu, Liping & Ma, Yanyuan, 2017. "Efficient dimension reduction for multivariate response data," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 187-199.
    18. Germán Aneiros & Philippe Vieu, 2015. "Partial linear modelling with multi-functional covariates," Computational Statistics, Springer, vol. 30(3), pages 647-671, September.
    19. B. Ettinger & S. Perotto & L. M. Sangalli, 2016. "Spatial regression models over two-dimensional manifolds," Biometrika, Biometrika Trust, vol. 103(1), pages 71-88.
    20. Sang, Peijun & Lockhart, Richard A. & Cao, Jiguo, 2018. "Sparse estimation for functional semiparametric additive models," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 105-118.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:bla:istatr:v:88:y:2020:i:1:p:142-154. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: https://edirc.repec.org/data/isiiinl.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.