The Annals of Statistics
- Ann. Statist.
- Volume 30, Number 5 (2002), 1225-1310.
What is a statistical model?
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.
Ann. Statist. Volume 30, Number 5 (2002), 1225-1310.
First available in Project Euclid: 28 October 2002
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Aggregation agricultural field experiment Bayes inference Box-Cox model category causal inference commutative diagram conformal model contingency table embedding exchangeability extendability extensive variable fertility effect functor Gibbs model harmonic model intensive variable interference Kolmogorov consistency lattice process measure process morphism natural parameterization natural subparameter opposite category quadratic exponential model representation spatial process spline model type III model
McCullagh, Peter. What is a statistical model?. Ann. Statist. 30 (2002), no. 5, 1225--1310. doi:10.1214/aos/1035844977. https://projecteuclid.org/euclid.aos/1035844977.
- Includes: Julian Besag. Comment.
- Includes: Peter J. Bickel. Comment.
- Includes: Hans Brøns. Comment.
- Includes: D. A. S. Fraser, N. Reid. Comment.
- Includes: Inge S. Helland. Comment.
- Includes: Peter J. Huber. Comment.
- Includes: Rudolf Kalman. Comment.
- Includes: Steve Pincus. Comment.
- Includes: Joe Tjur. Comment.
- Includes: Peter McCullagh. Rejoinder.