HISTORY OF RELATIONSHIPS 5
(2) the distribution of the errors is not affected by the grouping.
Note that (i) is a strict inequality.
It has sometimes been incorrectly thought that it suffices to group on the
basis of the observed
X
1
,
placing the
m
smallest
X
1
in one group and the
m
largest in the other. Condition (i) may still hold, providing certain re-
strictions are placed upon errors, but (ii) will certainly not hold for normally
distributed errors. In this situation, in an extreme case it is conceivable that
all observations with "negative" errors may fall in one group and all with
"positive" errors in the other. Various conditions have been determined for
Wald's requirements to hold. Similar, or even more stringent, restrictions ap-
ply for the validity of a nonparametric estimator proposed by Theil (1950).
The inherent difficulty in Theil's method (and several proposed modifica-
tions) is that ordering by true
x
and ordering by observed
X
will only be
certain to coincide if very stringent bounds are placed on errors.
The fitting of a linear relationship with errors in the variables is unusual
in statistical theory and practice in that normality assumptions are, generally
speaking, a complication. In the immediate post-World War II era, papers
by Lindley (1947), Geary (1949), Reiersel (1950), Neyman and Scott (1948,
1951), and Kiefer and Wolfowitz (1956) clarified many of the problems of
identifiability and consistency that dogged the estimation problem for one or
both of the models that Kendall ( 19 51, 19 52) distinguished as functional or
structural relationships.
Lindley's paper deals primarily with a regression problem where there is a
linear relationship between true unobservables of the form
(3)
E(yJx)
=Po+ P
1x,
and he obtained conditions on the error distribution that would preserve
linearity of regression for the observables, i.e., such that the regression for
observables would be of the form
(4)
Lindley's result was given in terms of cumulants and links indirectly to the
work of Geary (1949). Geary showed that in the absence of knowledge about
the error variance, estimation in a structural relationship was possible only
if cumulants of higher order than the second were nonzero. This is con-
sistent with work by Reiersel ( 1950), who showed that either nonnormality
of the
x ,
or of the errors, was needed for identifiability in the absence of
certain minimal knowledge about error variances. Geary's cumulant method
is analogous to the method of moments. It cannot be used for the normal
distribution, because cumulants of order higher than two vanish. Geary's
method seems to have been little used in practice, partly because there is an
arbitrary element in the choice of cumulants to be used, but more impor-
tantly because it is intuitively obvious that the closer the error structure is
to normality the greater the difficulty associated with high-order cumulants
being near to vanishing. Madansky ( 1959) compared a number of methods
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