Section: Statistics & Machine learning
Topic: Statistics

On Nonparanormal Likelihoods

Hothorn, Torsten1 orcid 
1Institut für Epidemiologie, Biostatistik und Prävention Universität Zürich Hirschengraben 84, CH-8001 Zürich, Switzerland

Corresponding author(s): Hothorn, Torsten (Torsten.Hothorn@R-project.org)

10.24072/pcjournal.814 - Peer Community Journal, Volume 6 (2026), article no. e105

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Nonparanormal models describe the joint distribution of multivariate responses via latent Gaussian, and thus parametric, copulae while allowing flexible nonparametric marginals. Some aspects of such distributions, for example conditional independence, are formulated parametrically. Other features, such as marginal distributions, can be formulated non- or semiparametrically. Such models are attractive when multivariate normality is questionable but interpretability paramount. Most estimation procedures perform two steps, first estimating the nonparametric part. The copula parameters come second, treating the marginal estimates as known. This is sufficient for some applications. For other applications, e.g. when a semiparametric margin features parameters of interest or when standard errors are important, a simultaneous estimation of all parameters might be more advantageous. We present suitable parameterisations of nonparanormal models, possibly including semiparametric effects, and define four novel nonparanormal log-likelihood functions. In general, the corresponding one-step optimisation problems are shown to be non-convex. In some cases, however, biconvex problems emerge. Several convex approximations are discussed. From a low-level computational point of view, the core contribution is the score function for multivariate normal log-probabilities computed via Genz’ procedure. As a demonstration for the versatility of the theoretical and computational framework, we present a series of nonparanormal models for transformation discriminant analysis when some biomarkers are subject to limit-of-detection problems. Possible empirical gains of full maximum likelihood estimation compared to two-step approaches are illustrated in a simulation study targeting semiparametric efficient polychoric correlation analysis where a theoretical benchmark is available.

Published online:
DOI: 10.24072/pcjournal.814
Type: Research article
Classification:
Keywords: Methodology (stat.ME), Computation (stat.CO), FOS: Computer and information sciences

Hothorn, Torsten  1

1 Institut für Epidemiologie, Biostatistik und Prävention Universität Zürich Hirschengraben 84, CH-8001 Zürich, Switzerland
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Hothorn, T. On Nonparanormal Likelihoods. Peer Community Journal, Volume 6 (2026), article  no. e105. https://doi.org/10.24072/pcjournal.814
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PCI peer reviews and recommendation, and links to data, scripts, code and supplementary information: 10.24072/pci.statml.100138

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The recommender in charge of the evaluation of the article and the reviewers declared that they have no conflict of interest (as defined in the code of conduct of PCI) with the authors or with the content of the article.

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