The preceding section outlined the statistical
model for **S** which
*SEPATH* attempts to fit to the sample
data. If the model fits perfectly in the population, then Equation 41
holds. Such perfection is, of course, extremely unlikely to happen.

It is, in fact, virtually certain that Equation
41 does *not* hold exactly for your statistical population, and that
in fact an additional error term *pop*
should be added to the right side of the equation. The size of the elements
of this error matrix would reflect how badly a particular model fits in
the population. You could find out what **E***pop**
*was if you somehow knew **S**.
(You would simply input **S**
to *SEPATH* and fit your model to it.)
If you did, you would be faced with a difficult problem of exactly how
to quantify the information in **E***pop*.

There is an additional complication. In practice,
you do not know **S**.
You only have **S**, an estimate of **S**
from sample data. It is this estimate, usually the ordinary sample covariance
matrix based on *N* independent observations, which one attempts
to fit with *SEPATH*.

Consequently, in practice one attempts to fit
**S** rather than **S**
with the model of Equation 41, and one obtains, as a result of this model
fitting procedure, a *sample* matrix of residuals **E***samp*.
In general the object of the estimation process is to make the elements
of **E***samp* as "small
as possible" in some sense. This notion of "smallness"
is quantified in a "discrepancy function."

General Properties of Discrepancy Functions

Maximum Wishart Likelihood Estimation

Iteratively Reweighted GLS Estimation