Statistical models and likelihood functions

1. Nov 17, 2012

Cinitiator

1. The problem statement, all variables and given/known data
I have a couple of notation interpretation questions:
1) What does $$f_X(x|θ)$$ represent in this case? The realization function of of our random vector X for some value x and a parameter θ (so that if our random vector has n random variables, its realization vector will be a subset of R^n)? Or is something else represented here?

2) If our (non-parametrized) statistical model is based on some random vector X with n random variables, will it contain realization functions of the random vector, or rather the random variable functions which the said vector contains?

3) In case of parametrized models: Is the statistical model set (let's name it P) a set of functions under every parameter space possible? And what do these functions represent? Are they assumed to have a fixed input? Are they realizations of a random vector under every single parameter in the parameter space? Or are they random variable functions which belong to our random vector?

2. Relevant equations

3. The attempt at a solution
Trying to interpret it in different ways, but not knowing which interpretation is the correct one.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Nov 17, 2012

haruspex

It is the probability (density?) function for the r.v. X given the observations θ.
I don't know what you mean by a realization function.

3. Nov 18, 2012

Cinitiator

I know that. I don't know in what precise way this function is 'generated'. That is - do we take an entire random vector (say, an R^n vector with random variables) and input it, and then output a R^n vector in the measure space (probabilities for each R^...)? That's what I call a realization function, since it outputs the realization of this random vector.

Or is the said function generated in an entirely different way given our random variable?

4. Nov 18, 2012

haruspex

No, it can't be that. If you look at the definition of the likelihood function you can deduce that the range of f is ℝ, not ℝn. So I would say it's just a joint distribution.