They recognize (page 319) that when the transformation is one-to-one the proof of the invariance property is very simple. But then they extend the invariance property to arbitrary transformations of the parameters introducing an induced likelihood function on page 320. Theorem 7.2.10 on the same page gives the proof of the extended property.
Invariance property of MLE: what is the MLE of $\theta^2$ of normal ...
Invariance vs. Equivariance Santanu_Pattanayak's answer (here) points out that there is a difference between translation invariance and translation equivariance. Translation invariance means that the system produces exactly the same response, regardless of how its input is shifted.
5 I understand that scalar invariance, in the context of Structural Equations Modeling (SEM), is having the intercepts for observed variables loading on the same latent variable be invariant across multiple groups. However - what does scalar invariance mean substantially? What are its implications?
I then ensured measurement invariance across these groups, achieving configural, metric, and scalar invariance. For the final analysis phase, I aim to compare the latent means of the two factors across these groups. For instance, I am interested in determining whether Factor 1 is statistically higher in Group 2 compared to Group 3.
Casella and Berger (2002) explain this by saying that, "In many cases, this simple version of the invariance of MLEs is not useful because many of the functions we are interested in are not one-to-one" (p. 320). This is the essence of their motivation for extending the concept of the likelihood function. If you have an existing sampling density framed in terms of a minimal sufficient parameter ...