Optimality

IMS Lecture Notes–Monograph Series
2nd Lehmann Symposium – Optimality
Vol. 49 (2006) 98–119
c○ Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000419
Where do statistical models come from?
Revisiting the problem of specification
Aris Spanos ∗1
Virginia Polytechnic Institute and State University
Abstract: R. A. Fisher founded modern statistical inference in 1922 and identified
its fundamental problems to be: specification, estimation and distribution.
Since then the problem of statistical model specification has received scant
attention in the statistics literature. The paper traces the history of statistical
model specification, focusing primarily on pioneers like Fisher, Neyman, and
more recently Lehmann and Cox, and attempts a synthesis of their views in the
context of the Probabilistic Reduction (PR) approach. As argued by Lehmann
[11], a major stumbling block for a general approach to statistical model specification
has been the delineation of the appropriate role for substantive subject
matter information. The PR approach demarcates the interrelated but complemenatry
roles of substantive and statistical information summarized ab initio
in the form of a structural and a statistical model, respectively. In an attempt
to preserve the integrity of both sources of information, as well as to ensure the
reliability of their fusing, a purely probabilistic construal of statistical models
is advocated. This probabilistic construal is then used to shed light on a
number of issues relating to specification, including the role of preliminary
data analysis, structural vs. statistical models, model specification vs. model
selection, statistical vs. substantive adequacy and model validation.
1. Introduction
The current approach to statistics, interpreted broadly as ‘probability-based data
modeling and inference’, has its roots going back to the early 19th century, but it
was given its current formulation by R. A. Fisher [5]. He identified the fundamental
problems of statistics to be: specification, estimation and distribution. Despite its
importance, the question of specification, ‘where do statistical models come from?’
received only scant attention in the statistics literature; see Lehmann [11].
The cornerstone of modern statistics is the notion of a statistical model whose
meaning and role have changed and evolved along with that of statistical modeling
itself over the last two centuries. Adopting a retrospective view, a statistical model
is defined to be an internally consistent set of probabilistic assumptions aiming to
provide an ‘idealized’ probabilistic description of the stochastic mechanism that
gave rise to the observed data x := (x1, x2, . . . , xn). The quintessential statistical
model is the simple Normal model, comprising a statistical Generating Mechanism
(GM):
(1.1) Xk = µ + uk, k∈ N :={1,2, . . . n, . . .}
∗ I’m most grateful to Erich Lehmann, Deborah G. Mayo, Javier Rojo and an anonymous
referee for valuable suggestions and comments on an earlier draft of the paper.
1 Department of Economics, Virginia Polytechnic Institute, and State University, Blacksburg,
VA 24061, e-mail: aris@vt.edu
AMS 2000 subject classifications: 62N-03, 62A01, 62J20, 60J65.
Keywords and phrases: specification, statistical induction, misspecification testing, respecification,
statistical adequacy, model validation, substantive vs. statistical information, structural vs.
statistical models.
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