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Sufficient dimension reduction and prediction in regression: Asymptotic results

Author

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  • Forzani, Liliana
  • Rodriguez, Daniela
  • Smucler, Ezequiel
  • Sued, Mariela

Abstract

We consider model-based sufficient dimension reduction for generalized linear models and prove the consistency and asymptotic normality of the prediction estimator studied empirically for the normal case by Adragni and Cook (2009) when a sample version of the sufficient dimension reduction is used. Moreover, we provide a formula for the prediction that does need require explicitly computing the reduction.

Suggested Citation

  • Forzani, Liliana & Rodriguez, Daniela & Smucler, Ezequiel & Sued, Mariela, 2019. "Sufficient dimension reduction and prediction in regression: Asymptotic results," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 339-349.
    • Handle: RePEc:eee:jmvana:v:171:y:2019:i:c:p:339-349
    DOI: 10.1016/j.jmva.2018.12.003
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    References listed on IDEAS

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    1. Cook, R. Dennis & Forzani, Liliana, 2009. "Likelihood-Based Sufficient Dimension Reduction," Journal of the American Statistical Association, American Statistical Association, vol. 104(485), pages 197-208.
    2. Efstathia Bura & Sabrina Duarte & Liliana Forzani, 2016. "Sufficient Reductions in Regressions With Exponential Family Inverse Predictors," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(515), pages 1313-1329, July.
    3. Zhu, Yu & Zeng, Peng, 2006. "Fourier Methods for Estimating the Central Subspace and the Central Mean Subspace in Regression," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1638-1651, December.
    4. Shizhe Chen & Daniela M. Witten & Ali Shojaie, 2015. "Selection and estimation for mixed graphical models," Biometrika, Biometrika Trust, vol. 102(1), pages 47-64.
    5. Li, Bing & Wang, Shaoli, 2007. "On Directional Regression for Dimension Reduction," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 997-1008, September.
    6. Yingcun Xia & Howell Tong & W. K. Li & Li‐Xing Zhu, 2002. "An adaptive estimation of dimension reduction space," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(3), pages 363-410, August.
    7. Efstathia Bura & R. Dennis Cook, 2001. "Estimating the structural dimension of regressions via parametric inverse regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 393-410.
    8. Izenman, Alan Julian, 1975. "Reduced-rank regression for the multivariate linear model," Journal of Multivariate Analysis, Elsevier, vol. 5(2), pages 248-264, June.
    9. Cook, R. Dennis & Ni, Liqiang, 2005. "Sufficient Dimension Reduction via Inverse Regression: A Minimum Discrepancy Approach," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 410-428, June.
    10. Jie Cheng & Elizaveta Levina & Pei Wang & Ji Zhu, 2014. "A sparse ising model with covariates," Biometrics, The International Biometric Society, vol. 70(4), pages 943-953, December.
    11. Diego Tomassi & Liliana Forzani & Efstathia Bura & Ruth Pfeiffer, 2017. "Sufficient dimension reduction for censored predictors," Biometrics, The International Biometric Society, vol. 73(1), pages 220-231, March.
    12. R. Dennis Cook & Liliana Forzani, 2008. "Covariance reducing models: An alternative to spectral modelling of covariance matrices," Biometrika, Biometrika Trust, vol. 95(4), pages 799-812.
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    Cited by:

    1. Baek, Seungchul & Hoyoung, Park & Park, Junyong, 2024. "Variable selection using data splitting and projection for principal fitted component models in high dimension," Computational Statistics & Data Analysis, Elsevier, vol. 196(C).

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