I have heard of Owen's empirical likelihood, but until recently paid it no heed until I came across it in a paper of interest (Mengersen et al. 2012).
In my efforts to understand it, I have gleaned that the likelihood of the observed data is represented as $$L = \prod_i p_i = \prod_i P(X_i=x) = \prod_i P(X_i \le x) - P(X_i \lt x)$$ , where $\sum_i p_i = 1$ and $p_i > 0$.
However, I have been unable to make the mental leap connecting this representation with how it can be used to make inferences about observations. Perhaps I am too rooted in thinking of a likelihood w.r.t parameters of a model?
Regardless, I have been searching Google Scholar for some paper employing empirical likelihood that would help me internalize the concept... to no avail. Obviously, there is Art Owen's book on Empirical Likelihood, but Google Books leaves out all the yummy bits and I'm still in the slow process of getting an inter-library loan.
In the meantime, can somebody kindly point me to papers and documents that clearly illustrate the premise of empirical likelihood and how it is employed? An illustrative description of EL itself would also be welcome!