Statistical Estimation - Chi-square Test Statistics

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


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.