profile likelihood

The profile likelihood (Lee, Nelder and Pawitan, 2007) Definition: Given the joint likelihood L(delta, eta) the profile likelihood of delta is given by L(delta)=max@eta L(delta, eta), where the maximization is performed at a fixed value of delta. The last sentence means that likelihood of delta will be hit the peak at a fixed value of delta in the possible domain no matter what values the eta will have. Also, given delta, the MLE of eta is a function of delta, L(delta)=L(delta, eta-hat_@detal) An interval can be obtained based upon the likelihood function. The interval contains the parameters with largest likelihoods. If there are nuisance parametes, the profile likelihood method can be used to “control” the effect of them. Take the HGLM (GLMMs) as a example