Difference between restricted and unrestricted parameter space in MLE

Question

I searched on the internet but I could not find any clues about my question. Can anyone just simply tell what is the difference between restricted and unrestricted parameter space in MLE?

I have used bimodalitytest package in R, the function bimodality.test which performs the likelihood ratio test for bimodality.

It's description is:

"This function performs the likelihood ratio test for a given dataset. It tests the null hypothesis, whether a two components normal mixture is bimodal. Therefore it calculates the maximum likelihood estimators for the restricted and non restricted parameter space and returns for example the likelihoodratio and the p-value."

And I know the next question depends on the purpose of research and other points, but let me also ask it:

Which one do you suggest for MLE? restricted or non restricted parameter space? On which basis should I decide on using them?

Answer 1

If you read the available theory referenced in the help page, you see that the authors are building a likelihood for a mixture model and considering a dividing line (in the case of two component models) between the regions of the parameter space where there are two modes and only one mode. The "restricted likelihood" is estimated on the region where the distribution is unimodal and the unrestricted likelihood is the estimated on the full set of parameters.

Not all two-component mixture models are bi-modal. They consider models of the form:

 f(x;θ1,θ2,p) = pf(x;θ1) + (1 − p)f(x;θ2), where 

               f(x;θ1,θ2,p) = pf(x;θ1) + (1 − p)f(x;θ2),
             (θ1,θ2,p) ∈ Θ × Θ × [0,1] = Θ[mix] ⊂ R2d+1.

Obviously the case where p is 0 or 1 is unimodal, but there are also values of thetas were unimodality occurs and they develop a theory that establishes what might be called "inferential continuity" across the boundary between unimodal regions and bimodal regions in a variety of situations for both Normal and von Mises distributions.