What is a statistical model

  title={What is a statistical model},
  author={Peter McCullagh},
  journal={Annals of Statistics},
  • P. McCullagh
  • Published 1 October 2002
  • Philosophy
  • Annals of Statistics
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… 
This paper illustrates and exemplifies that a number of issues arise from decisions to choose a parametric statistical model, even in relatively simple settings, such as ordinal regression, linear mixed models, models for cross-classified data and generalizedlinear mixed models.
Validity and the foundations of statistical inference
A demonstration that the inferential model framework meets the proposed criteria for valid and prior-free statistical inference, thereby solving perhaps the most important unsolved problem in statistics.
A sketch of category theoretic interface for experiment-theory relationship
What is the meaning of the terms experiment and experimental verification? Can they be formalised mathematically? Consideration of an experiment as something exhaustively specified by a single set X
A theory of statistical models for Monte Carlo integration
Summary. The task of estimating an integral by Monte Carlo methods is formulated as a statistical model using simulated observations as data. The difficulty in this exercise is that we ordinarily
Three enigmatic examples and inference from likelihood
Statistics has many inference procedures for examining a model with data to obtain information concerning the value of a parameter of interest. If these give different results for the same model and
On Some Principles of Statistical Inference
Some principles of statistical inference are discussed, to outline how these are, or could be, used to inform the interpretation of results, and to provide a greater degree of coherence for the foundations of statistics.
The algebra and machine representation of statistical models
This dissertation takes steps toward digitizing and systematizing two major artifacts of data science, statistical models and data analyses, by designing and implementing a software system for creating machine representations of data analyses in the form of Python or R programs.
Identified Parameters , Parameters of Interest and Their Relationships
The goal of this note is to provide some additional results to the interesting and provocative paper of Maris and Bechger (2009). More specifically, we have three aims. First, we want to distinguish


Conditional inference and Cauchy models
SUMMARY Many computations associated with the two-parameter Cauchy model are shown to be greatly simplified if the parameter space is represented by the complex plane rather than the real plane. With
Theory of Probability.
DR. KEYNES'S book is a searching analysis of the fundamental principles of the theory of probability and of the particular judgments involved in its application to concrete problems. He adopts the
Quantum Theory from Symmetries in a General Statistical Parameter Space
The aim of this paper is to show a connection between an extended theory of statistical experiments on the one hand and the foundation of quantum theory on the other hand. The main aspects of this
An Analysis of Transformations Revisited, Rebutted
Abstract : In a paper written in 1964, Box and Cox described a method for estimating transformations and showed how in suitable cases valuable increases in simplicity and efficiency were possible.
Nonlinear Regression, Quasi Likelihood, and Overdispersion in Generalized Linear Models
Abstract The aim of this article is to reconsider the methods for handling of overdispersion in generalized linear models proposed by McCullagh and Nelder. Our starting point will be a nonlinear
Quantum Mechanics from Symmetry and Statistical Modeling
The derivation relies partly on quantumlogic, partly on a group representation of the hyperparameter group, where the invariant spaces are shown to be in 1-1 correspondence with the equivalenceclasses of permissible parametric functions.
An Analysis of Transformations Revisited
Abstract Following Box and Cox (1964), we assume that a transform Z i = h(Yi , λ) of our original data {Yi } satisfies a linear model. Consistency properties of the Box-Cox estimates (MLE's) of λ and
Marginalization Paradoxes in Bayesian and Structural Inference
We describe a range of routine statistical problems in which marginal posterior distributions derived from improper prior measures are found to have an unBayesian property-one that could not occur if
Not all (possibly) "random" sequences are created equal.
  • S. Pincus, R. Kálmán
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1997
This work assess randomness via approximate entropy (ApEn), a computable measure of sequential irregularity, applicable to single sequences of both finite and infinite length, and indicates the novelty and facility of the multidimensional viewpoint taken by ApEn, in contrast to classical measures.
Quotient spaces and statistical models
The purpose of this paper is to draw attention to the widespread occurrence of quotient spaces in statistical work. Quotient spaces are intrinsic to probability distributions, residuals and