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October 2002 What is a statistical model?
Peter McCullagh
Ann. Statist. 30(5): 1225-1310 (October 2002). DOI: 10.1214/aos/1035844977

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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Peter McCullagh. "What is a statistical model?." Ann. Statist. 30 (5) 1225 - 1310, October 2002. https://doi.org/10.1214/aos/1035844977

Information

Published: October 2002
First available in Project Euclid: 28 October 2002

zbMATH: 1039.62003
MathSciNet: MR1936320
Digital Object Identifier: 10.1214/aos/1035844977

Subjects:
Primary: 62AO5
Secondary: 62F99

Rights: Copyright © 2002 Institute of Mathematical Statistics

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Vol.30 • No. 5 • October 2002
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