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An informative prior distribution on functions with application to functional regression

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  • Christophe Abraham

Abstract

We provide a prior distribution for a functional parameter so that its trajectories are smooth and vanish on a given subset. This distribution can be interpreted as the distribution of an initial Gaussian process conditioned to be zero on a given subset. Precisely, we show that the initial Gaussian process is the sum of the conditioned process and an independent process with probability one and that all the processes have the same almost sure regularity. This prior distribution is use to provide an interpretable estimate of the coefficient function in the linear scalar‐on‐function regression; by interpretable, we mean a smooth function that may possibly be zero on some intervals. We apply our model in a simulation and real case studies with two different priors for the null region of the coefficient function. In one case, the null region is known to be an unknown single interval. In the other case, it can be any unknown unions of intervals.

Suggested Citation

  • Christophe Abraham, 2024. "An informative prior distribution on functions with application to functional regression," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 78(2), pages 357-373, May.
    • Handle: RePEc:bla:stanee:v:78:y:2024:i:2:p:357-373
    DOI: 10.1111/stan.12322
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    References listed on IDEAS

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    1. Robert Tibshirani & Michael Saunders & Saharon Rosset & Ji Zhu & Keith Knight, 2005. "Sparsity and smoothness via the fused lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 91-108, February.
    2. Philip T. Reiss & Jeff Goldsmith & Han Lin Shang & R. Todd Ogden, 2017. "Methods for Scalar-on-Function Regression," International Statistical Review, International Statistical Institute, vol. 85(2), pages 228-249, August.
    3. Zhang, Hao, 2004. "Inconsistent Estimation and Asymptotically Equal Interpolations in Model-Based Geostatistics," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 250-261, January.
    4. Crainiceanu, Ciprian M. & Goldsmith, A. Jeffrey, 2010. "Bayesian Functional Data Analysis Using WinBUGS," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 32(i11).
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