In the context of covariance structure models (as used in SEM), we have $\Sigma$ a population covariance structure, and $\Sigma(\cdot)$, a function of a parameter vector that returns a model-implied covariance structure. Then, a discrepancy function $F(\Sigma, \Sigma(\cdot))$, also called objective function or fit function, is a real valued function that fulfills the following criteria:
- $F(\Sigma, \Sigma(\cdot)) \geq 0$ for every $\Sigma$ and $\Sigma(\cdot)$.
- $F(\Sigma, \Sigma(\cdot)) = 0$ if and only if $\Sigma =\Sigma(\cdot)$.
- Is twice continuously differentiable with respect to $\Sigma$ and $\Sigma(\cdot)$.
Considering the above, why is monotonicity not a requirement of the function?
A plausible answer I can think of is that we only care about optimizing this function, thus we do not restrict its functional form. But if we have no interest in its intuitive interpretation as a measure of discrepancy (i.e. a higher value is a higher discrepancy from the target structure), why even bother with the first criterion?