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Cited by 248Statistical Methods and Bayesian InferenceProbability and Statistical ResearchStatistical Methods and Applications

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?

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McCullagh, Peter·2002·The Annals of Statistics
DOI 10.1214/aos/1035844977
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Abstract

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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Research works printed but not linked: 15

  1. Billingsley, P. (1986). Probability and Measure, 2nd ed. Wiley, New York.
  2. MCCULLAGH, P. (2000). Invariance and factorial models (with discussion). J. Roy. Statist. Soc. Ser. B 62 209-256. · DOI
  3. GELMAN, A., CARLIN, J. B., STERN, H. and RUBIN, D. B. (1995). Bayesian Data Analy sis. Chapman and Hall, London. · DOI · Paper page
  4. MAC LANE, S. (1998). Categories for the Working Mathematician, 2nd ed. Springer, New York.
  5. MCCULLAGH, P. (1999). Quotient spaces and statistical models. Canad. J. Statist. 27 447-456. · DOI
  6. WICHURA, M. (2001). Some de Finetti ty pe theorems. Preprint.
  7. DE FINETTI, B. (1975). Theory of Probability 2. Wiley, New York.
  8. HELLAND, I. S. (1999a). Quantum mechanics from sy mmetry and statistical modelling. Internat. J. Theoret. Phy s. 38 1851-1881. · DOI
  9. MCCULLAGH, P. (1980). Regression models for ordinal data (with discussion). J. Roy. Statist. Soc. Ser. B 42 109-142. · DOI · Paper page
  10. TJUR, T. (2000). Contribution to the discussion of McCullagh (2000). J. Roy. Statist. Soc. Ser. B 62 238-239. · DOI
  11. BESAG, J. E. (1974). Spatial interaction and the statistical analysis of lattice sy stems (with discussion). J. Roy. Statist. Soc. Ser. B 36 192-236. · DOI · Paper page
  12. BESAG, J. E. and HIGDON, D. M. (1999). Bayesian analysis of agricultural field experiments (with discussion). J. Roy. Statist. Soc. Ser. B 61 691-746. · DOI · Paper page
  13. BESAG, J. E. and KOOPERBERG, C. L. (1995). On conditional and intrinsic autoregressions. Biometrika 82 733-746. · DOI · Paper page
  14. FRASER, D. A. S. (1968b). The Structure of Inference. Wiley, New York.
  15. MACCULLAGH, J. (1839). An essay towards the dy namical theory of cry stalline reflexion and refraction. Trans. Roy. Irish Academy 21 17-50.

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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
DOI
10.1214/aos/1035844977
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
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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