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

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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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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: 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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