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Partial least squares estimator for single‐index models

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  • Prasad Naik
  • Chih‐Ling Tsai

Abstract

The partial least squares (PLS) approach first constructs new explanatory variables, known as factors (or components), which are linear combinations of available predictor variables. A small subset of these factors is then chosen and retained for prediction. We study the performance of PLS in estimating single‐index models, especially when the predictor variables exhibit high collinearity. We show that PLS estimates are consistent up to a constant of proportionality. We present three simulation studies that compare the performance of PLS in estimating single‐index models with that of sliced inverse regression (SIR). In the first two studies, we find that PLS performs better than SIR when collinearity exists. In the third study, we learn that PLS performs well even when there are multiple dependent variables, the link function is non‐linear and the shape of the functional form is not known.

Suggested Citation

  • Prasad Naik & Chih‐Ling Tsai, 2000. "Partial least squares estimator for single‐index models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(4), pages 763-771.
    • Handle: RePEc:bla:jorssb:v:62:y:2000:i:4:p:763-771
    DOI: 10.1111/1467-9868.00262
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    Cited by:

    1. Li‐Ping Zhu & Li‐Xing Zhu, 2009. "On distribution‐weighted partial least squares with diverging number of highly correlated predictors," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 525-548, April.
    2. Dominic S.K. Lim & Eric A. Morse & Ronald K. Mitchell & Kristie K. Seawright, 2010. "Institutional Environment and Entrepreneurial Cognitions: A Comparative Business Systems Perspective," Entrepreneurship Theory and Practice, , vol. 34(3), pages 491-516, May.
    3. Hyonho Chun & Sündüz Keleş, 2010. "Sparse partial least squares regression for simultaneous dimension reduction and variable selection," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(1), pages 3-25, January.
    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. Qingming Zou & Zhongyi Zhu & Jinglong Wang, 2009. "Local influence analysis for penalized Gaussian likelihood estimation in partially linear single-index models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(4), pages 905-918, December.
    6. Hansheng Wang & Chih‐Ling Tsai, 2009. "‘Model selection for generalized linear models with factor‐augmented predictors’," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 25(3), pages 241-242, May.
    7. Ghosh, Debashis, 2011. "Propensity score modelling in observational studies using dimension reduction methods," Statistics & Probability Letters, Elsevier, vol. 81(7), pages 813-820, July.
    8. Martin, Manon & Govaerts, Bernadette, 2019. "Feature Selection in metabolomics with PLS-derived methods," LIDAM Discussion Papers ISBA 2019020, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    9. Philip Hans Franses & Rianne Legerstee, 2010. "A Unifying View On Multi‐Step Forecasting Using An Autoregression," Journal of Economic Surveys, Wiley Blackwell, vol. 24(3), pages 389-401, July.
    10. Matthew F. Dixon & Nicholas G. Polson & Kemen Goicoechea, 2022. "Deep Partial Least Squares for Empirical Asset Pricing," Papers 2206.10014, arXiv.org.
    11. Qingming Zou & Zhongyi Zhu, 2014. "M-estimators for single-index model using B-spline," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(2), pages 225-246, February.
    12. Qin Wang & Yuan Xue, 2023. "A structured covariance ensemble for sufficient dimension reduction," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 17(3), pages 777-800, September.
    13. Nguyen Tuan S & Rojo Javier, 2009. "Dimension Reduction of Microarray Data in the Presence of a Censored Survival Response: A Simulation Study," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 8(1), pages 1-38, January.
    14. Lai, Peng & Wang, Qihua & Lian, Heng, 2012. "Bias-corrected GEE estimation and smooth-threshold GEE variable selection for single-index models with clustered data," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 422-432.
    15. Naik, Prasad A. & Tsai, Chih-Ling, 2004. "Isotonic single-index model for high-dimensional database marketing," Computational Statistics & Data Analysis, Elsevier, vol. 47(4), pages 775-790, November.
    16. Bousebata, Meryem & Enjolras, Geoffroy & Girard, Stéphane, 2023. "Extreme partial least-squares," Journal of Multivariate Analysis, Elsevier, vol. 194(C).
    17. Ceren Kolsarici & Demetrios Vakratsas, 2015. "Correcting for Misspecification in Parameter Dynamics to Improve Forecast Accuracy with Adaptively Estimated Models," Management Science, INFORMS, vol. 61(10), pages 2495-2513, October.
    18. Dhiman, Neeraj & Jamwal, Mohit & Kumar, Ajay, 2023. "Enhancing value in customer journey by considering the (ad)option of artificial intelligence tools," Journal of Business Research, Elsevier, vol. 167(C).
    19. Pang, Zhen & Xue, Liugen, 2012. "Estimation for the single-index models with random effects," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 1837-1853.
    20. Han, Jinyue & Wang, Jun & Gao, Wei & Tang, Man-Lai, 2023. "Estimation of the directions for unknown parameters in semiparametric models," MPRA Paper 116365, University Library of Munich, Germany.

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