If **S** has a Wishart distribution, the
model is identified, and **q**
has *t* free parameters, then under fairly general conditions (*N*
- 1)FML
(**S**, **S(q)**),

(*N* - 1)FGLS
(**S**, **S(q)**),
and (*N* - 1)FIRGLS (**S**, **S(q)**),
all have an asymptotic *Chi-square* distribution with *p(p + 1)/2
- t* degrees of freedom.

Such a *Chi-square* statistic, often described
as a "goodness-of-fit" statistic (but perhaps more accurately
called a "badness-of-fit" statistic) allows us to test statistically
whether a particular model fits **S**
*perfectly* in the population (i.e., whether **S
= S(q)**).** **There is a long tradition of performing such
a test, although it is becoming increasingly clear that the procedure
is seldom appropriate.

Browne (1974) showed that, under typical assumptions
for maximum likelihood estimation, the two statistics (*N* - 1)FML
and (*N* - 1)F*ML, where

(60)

will converge stochastically, and will both
be distributed as *Chi-square* variates as *N* ® ∞ . Moreover, as *N* ® ∞ , the probability that the two discrepancy functions
FML and F*ML
will be minimized by different **q
**vectors converges to zero.

In practice, then, there tend to be only trivial
differences, if any, between estimates which minimize FML
and those which minimize F*ML.
This suggests that there will seldom be differences in practice between
**q **which minimize FML and those which minimize FIRGLS*. *As mentioned above,
Bentler (1989) has stated that the FIRGLS
and FML are equivalent methods,
i.e., lead to the same **q. **