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