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Distribution-Free Prediction Sets

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  • Jing Lei
  • James Robins
  • Larry Wasserman

Abstract

This article introduces a new approach to prediction by bringing together two different nonparametric ideas: distribution-free inference and nonparametric smoothing. Specifically, we consider the problem of constructing nonparametric tolerance/prediction sets. We start from the general conformal prediction approach, and we use a kernel density estimator as a measure of agreement between a sample point and the underlying distribution. The resulting prediction set is shown to be closely related to plug-in density level sets with carefully chosen cutoff values. Under standard smoothness conditions, we get an asymptotic efficiency result that is near optimal for a wide range of function classes. But the coverage is guaranteed whether or not the smoothness conditions hold and regardless of the sample size. The performance of our method is investigated through simulation studies and illustrated in a real data example.

Suggested Citation

  • Jing Lei & James Robins & Larry Wasserman, 2013. "Distribution-Free Prediction Sets," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 108(501), pages 278-287, March.
    • Handle: RePEc:taf:jnlasa:v:108:y:2013:i:501:p:278-287
    DOI: 10.1080/01621459.2012.751873
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    Citations

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    Cited by:

    1. Victor Chernozhukov & Kaspar Wüthrich & Yinchu Zhu, 2021. "An Exact and Robust Conformal Inference Method for Counterfactual and Synthetic Controls," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 116(536), pages 1849-1864, October.
    2. Ben O’Neill, 2022. "Smallest covering regions and highest density regions for discrete distributions," Computational Statistics, Springer, vol. 37(3), pages 1229-1254, July.
    3. Leying Guan & Robert Tibshirani, 2022. "Prediction and outlier detection in classification problems," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(2), pages 524-546, April.
    4. Philippe Goulet Coulombe & Mikael Frenette & Karin Klieber, 2023. "From Reactive to Proactive Volatility Modeling with Hemisphere Neural Networks," Papers 2311.16333, arXiv.org.
    5. Hu, Jianming & Luo, Qingxi & Tang, Jingwei & Heng, Jiani & Deng, Yuwen, 2022. "Conformalized temporal convolutional quantile regression networks for wind power interval forecasting," Energy, Elsevier, vol. 248(C).
    6. Victor Chernozhukov & Kaspar Wuthrich & Yinchu Zhu, 2019. "Distributional conformal prediction," Papers 1909.07889, arXiv.org, revised Aug 2021.
    7. Victor Chernozhukov & Kaspar Wüthrich & Yinchu Zhu, 2018. "Exact and robust conformal inference methods for predictive machine learning with dependent data," CeMMAP working papers CWP16/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    8. Xie, Haihan & Kong, Linglong, 2023. "Gaussian copula function-on-scalar regression in reproducing kernel Hilbert space," Journal of Multivariate Analysis, Elsevier, vol. 198(C).
    9. Mammen, Enno & Martínez Miranda, María Dolores & Nielsen, Jens Perch, 2015. "In-sample forecasting applied to reserving and mesothelioma mortality," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 76-86.
    10. Matteo Fontana & Gianluca Zeni & Simone Vantini, 2020. "Conformal Prediction: a Unified Review of Theory and New Challenges," Papers 2005.07972, arXiv.org, revised Jul 2022.
    11. Philippe Goulet Coulombe & Mikael Frenette & Karin Klieber, 2023. "From Reactive to Proactive Volatility Modeling with Hemisphere Neural Networks," Working Papers 23-04, Chair in macroeconomics and forecasting, University of Quebec in Montreal's School of Management, revised Nov 2023.
    12. Algo Carè & Simone Garatti & Marco C. Campi, 2017. "A coverage theory for least squares," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(5), pages 1367-1389, November.
    13. Diquigiovanni, Jacopo & Fontana, Matteo & Vantini, Simone, 2022. "Conformal prediction bands for multivariate functional data," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    14. David J. Olive, 2018. "Applications of hyperellipsoidal prediction regions," Statistical Papers, Springer, vol. 59(3), pages 913-931, September.
    15. João A. Bastos, 2023. "Conformal prediction of option prices," Working Papers REM 2023/0304, ISEG - Lisbon School of Economics and Management, REM, Universidade de Lisboa.

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