6
PETER SPRENT
of estimating slope of a bivariate relationship for a real data set, and the
method of cumulants behaved least well among all those considered.
While the regression content of Lindley's paper inrluded useful practical
results, his consideration of what is now called the functional relationship
model with normally distributed errors is of greater interest to us. He re-
garded the problem as one of estimating coefficients in the special case of the
regression model (3) where the conditional distribution of
y
given x is a
constant (i.e., has zero variance). In effect, (3) reduces to the form (1). In
passing, we note that the relationship for observed variables can equally well
be regarded as a generalization of the classical regression model where we
estimate
E(y1
jx1)
by
r;
when
x
1
is known, to a model in which
x
1
is itself
estimated by
X
1

Lindley assumed we observed
X
1
=
x
1
+u1 and
Y1
=
Y1
+e1
where
U1
,
e1
are errors that are NI[(O, 0), diag(auu'
aee)],
but he allowed
x1
to be mathematical variables or unknown constants (i.e., he did not assign
a distribution to them). Formal application of maximum likelihood estima-
tion (MLE) gave the estimator (2) and also the extraordinary relationship
between estimators
(5)
Lindley concluded this represented a failure of maximum likelihood as a
method of estimation. It was not until Solari ( 1969) showed that the solu-
tion represented a saddle point, and not a maximum, that a sound reason was
established for the apparent anomaly. If the ratio of the error variances is
assumed known, Lindley showed the ML estimator of slope is the now well-
known generalized least squares estimator. (See, e.g., Fuller ( 1987), equation
1.3. 7). There remained problems of consistency for estimators of the error
variances. These are explicable in terms of results of Kiefer and Wolfowitz
( 1956) on consistency of ML estimators when the number of parameters in-
creases with the number of observations, for here there are
n
+
4 parameters
for
n
observed points (since the
x
1
are themselves unknown nuisance pa-
rameters). These difficulties may be overcome in various ways. One is by the
use of pivotals - a technique proposed by many researchers in this and other
contexts involving nuisance parameters. Also, Morton ( 1981) introduced the
concept of unbiased estimating equations to avoid these difficulties.
3. Functional and structural relationships
The distinction between
functional
and
structural
relationships was first
clearly stated by Kendall ( 1951, 1952). In functional relationships the un-
derlying unobservables x in ( 1) are constants or "mathematical" variables
(without specific distributional properties). In
structural relationships
the
x
have a specified distribution, usually assumed normal, with parameters to
be estimated from the data. While in many cases it may be difficult to de-
cide whether a functional or structural model is appropriate for a particular
data set, in other cases it may be very clear that only one model is suitable.
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