Define **q**
as the current vector of free parameter values. Let **S**(**q**) represent a function which
models **S** as a function
of the *t* free parameter values in **q**.
The traditional approach to statistical estimation states as a model that

H0: **S
= S**(**q**) (47)

**S**(**q**) is assumed in our general
discussion to be any twice differentiable function of **q**.
In practice, it is usually restricted to the particular form of the general
model supported by the covariance structure software used for fitting
the model to data. For example, when fitting covariance matrices, *SEPATH*
is restricted to the model of Equation 41. In this case, assuming **B**, **G**,
and **X** have elements
which are either fixed numerical values or elements of**
q**, one may write

**S**(**q**) = **G**(**B**
- **I**)-1**GXG¢ **(**B¢ **-
**I**)-1**G¢ ** (48)

The discrepancy function F(**S**, **S**(**q**))
is a measure on **S** and **S**(**q**). In general, if a model
is identified (see *Model Identification* later in this section),
minimization of a discrepancy function satisfying the following three
restrictions will lead to consistent estimates for the elements of** q**:

F(**S**, **S**(**q**)) ³ 0

F(**S**, **S**(**q**)) = 0 if and only if **S**
= **S**(**q**).

F(**S**, **S**(**q**)) is continuous in **S**
and **S**(**q**).

The above notation, which is employed in many
books and papers on structural equation modeling, can be quite confusing
in practice, because **q **may
stand for different quantities in different situations. For example, when
we refer above to the discrepancy function F(**S**, **S**(**q**)), we are referring to *any
*permissible set of numbers employed as parameters in a model. In other
contexts, the values in **q **may
acquire a more specific meaning. For example, when we are referring to
the outcome of a maximum likelihood minimization process in which the
maximum likelihood discrepancy function has been minimized as a function
of **q**,**
**the elements of **q
**are now "maximum likelihood estimates."

Besides the *sample discrepancy function
*

F(**S**, **S**(**q**)), we may also discuss the
*population discrepancy function *F(**S**,
**S**(**q**)),
which we would obtain if we somehow knew **S**,
the population covariance matrix, and used our estimation algorithm to
fit the structural model to **S
**rather than **S**. We may write

(49)

Thus, the null hypothesis in Equation 47 may be expressed in several equivalent forms. For example,

H0: F(**S**,
**S**(**q**))
= 0 (50)

or

H0: (51)

As a simple consequence of the preceding definitions,
we can see that, when **q **is
*identified* (see *Model
Identification*), and the null hypothesis is true, **q
**is uniquely defined for *any *discrepancy function. However,
suppose the null hypothesis is *not* true, which under most conditions
is the reasonable assumption. In this case, we might define the "population
parameters" as those we would obtain if we somehow knew **S**, and fit a model to **S **by minimizing a discrepancy
function. The parameters in **q
**would then be defined as those that "fit best in the population."
The subtle problem here is that different discrepancy functions will usually
produce different **q **values.
Hence, although the point is hardly ever discussed in the literature,
**q** is, in practice,
hardly ever uniquely defined, unless you choose a *particular* discrepancy
function (say, maximum likelihood) as your criterion for choosing **q **"in the population."
The problem is that discrepancy functions have been chosen primarily on
the basis of their optimality properties for fitting **S** to a model,
*not* for fitting **S**.** **The reader should keep that
subtle point in mind when reading the following discussion of discrepancy
functions.