The DOI matches its Crossref record; 110 of the 110 references it cites were found in a registry or an index.
What is a statistical model?
Open accessAbstract
This paper addresses two closely related questions, "What is a statistical model?" and "What is a parameter?" The notions that a model must "make sense," and that a parameter must "have a well-defined meaning" are deeply ingrained in applied statistical work, reasonably well understood at an instinctive level, but absent from most formal theories of modelling and inference. In this paper, these concepts are defined in algebraic terms, using morphisms, functors and natural transformations. It is argued that inference on the basis of a model is not possible unless the model admits a natural extension that includes the domain for which inference is required. For example, prediction requires that the domain include all future units, subjects or time points. Although it is usually not made explicit, every sensible statistical model admits such an extension. Examples are given to show why such an extension is necessary and why a formal theory is required. In the definition of a subparameter, it is shown that certain parameter functions are natural and others are not. Inference is meaningful only for natural parameters. This distinction has important consequences for the construction of prior distributions and also helps to resolve a controversy concerning the Box-Cox model.
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“We argue that, as suggested in [46], it has a role to play in the study of point cloud data as well, and we give two illustrations of how this could happen, within the context of clustering.”
From Topology and data (2009) · cites this paper as [46]
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“tion of errors in his observations, and about facts that are assumed to be known for the purposes of statistical calculations. |[Colquhoun, 1971, p. v] Statistical models can take a variety of forms [McCullagh, 2002], but the model for the familiar Student’s t-test for independent samples is reasonably representative.…”
Background
Metadata
- Authors
- McCullagh, P.
- Affiliations
- University of Chicago (US)
- Corresponding author
- McCullagh, Peter
- ORCID
- McCullagh, Peter
- Venue
- The Annals of Statistics
- Publisher
- Institute of Mathematical Statistics
- Published
- 2002-10-01
- Volume
- 30
- Issue
- 5
- Pages
- 1225-1310
- ISSN
- 0090-5364 (ISSN-L)
- 2168-8966
- OpenAlex
- W2049954094
- Cited by
- 248
- References
- 103
- Citations by year
- 2026: 9 · 2025: 22 · 2024: 13 · 2023: 5 · 2022: 10 · 2021: 23 · 2020: 17 · 2019: 16 · 2018: 18 · 2017: 12 · 2016: 12 · 2015: 16 · 2014: 12 · 2013: 10 · 2012: 7
- FWCI
- 5.37
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- The Annals of Statistics (publishedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- CBS Research Portal (Copenhagen Business School) (submittedVersion)
- Funders
- National Science Foundation (0071726, 9705347)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Physical Sciences › Mathematics › Statistics and Probability › Probability and Statistical Research
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Applications
- Keywords
- statistical models, parameters, natural parameters, morphisms, functors, natural transformation, model extension, prediction, prior distributions
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