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.