Likelihood-based surrogate dimension reduction

Linh H. Nghiem*, Francis K. C. Hui, Samuel Muller, A. H. Welsh

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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Abstract

We consider the problem of surrogate sufficient dimension reduction, that is, estimating the central subspace of a regression model, when the covariates are contaminated by measurement error. When no measurement error is present, a likelihood-based dimension reduction method that relies on maximizing the likelihood of a Gaussian inverse regression model on the Grassmann manifold is well-known to have superior performance to traditional inverse moment methods. We propose two likelihood-based estimators for the central subspace in measurement error settings, which make different adjustments to the observed surrogates. Both estimators are computed based on maximizing objective functions on the Grassmann manifold and are shown to consistently recover the true central subspace. When the central subspace is assumed to depend on only a few covariates, we further propose to augment the likelihood function with a penalty term that induces sparsity on the Grassmann manifold to obtain sparse estimators. The resulting objective function has a closed-form Riemann gradient which facilitates efficient computation of the penalized estimator. We leverage the state-of-the-art trust region algorithm on the Grassmann manifold to compute the proposed estimators efficiently. Simulation studies and a data application demonstrate the proposed likelihood-based estimators perform better than inverse moment-based estimators in terms of both estimation and variable selection accuracy.

Original languageEnglish
Article number51
Pages (from-to)1-15
Number of pages15
JournalStatistics and Computing
Volume34
Issue number1
DOIs
Publication statusPublished - Feb 2024

Bibliographical note

Copyright © 2023, The Author(s). Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.

Keywords

  • Grassmann manifold
  • Measurement errors
  • Variable selection

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