Contemporary Mathematics
Volume 112, 1990
Some History of Functional
and Structural Relationships
PETER SPRENT
ABSTRACT. A serious study of statistical methods for estimating relation-
ships between variables subject to measurement and other errors began little
more than a century ago, but only slow progress was made before World War
II. Single linear relationships dominated theoretical developments up to that
time, although economists were already making practical use of simultaneous
relationship models, as were psychologists under the guise of factor analysis;
the last quarter century has seen an integration of developments in all these
areas as
well as an expansion to nonlinear errors-in-variable models. The
distinction between functional and structural relationships was formalized
by M. G. Kendall and led to a better understanding of many of the under-
lying problems. Measurement error models are now central to many studies
in this area
1. Early developments
Adcock (1877, 1878) is usually regarded as the first person to have con-
sidered seriously the problem of fitting a straight-line relationship when both
variables are subject to error, though Gauss clearly had measurement errors
in mind when he proposed the method of least squares. In essence, Ad-
cock obtained the least squares solution to a special case of the problem of
estimating
P
1
in a linear relationship
(1)
where, in the notation of Fuller ( 1987), the observables are
X
1
=
x
1
+
u1
and
Y1
=
y
1
+
e1 where u1
,
e1 are errors that are NI[(O, 0), diag(auu,
aee)].
Adcock assumed
auu
=
aee.
Kummel (1879) extended the result to the case
where the ratio of the error variances is supposed known.
1980 Mathematics Subject Classification (1985 Revision). OIA60, 62-03, 62J99.
Key words and phrases. Errors-in-equations, errors-in-variables, functional relationships,
measurement error models, regression, structural relationships ..
This paper is in final form, and no version of it will be submitted for publication elsewhere.
Financial support for travel from the Royal Society of Edinburgh and the American Mathe-
matical Society is gratefully acknowledged.
3
©
1990 American Mathematical Society
0271-4132/90 $1.00
+
$.25 per page
http://dx.doi.org/10.1090/conm/112/1087095
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