Let $\theta = (\theta_1, \theta_2, \dots)$ be a vector of parameters, and let $L(\theta) = L(\theta|y)$ denote the likelihood function with respect to observed data $y$.

Following the notation from here, suppose $\delta = \theta_j$ is the parameter of interest, and let $\xi$ denote the other parameters, i.e. $\xi = \theta \setminus \delta$. Then, the profile likelihood $L_p(\delta)$ takes the following expression:

$L_p(\delta) = \max_\xi L(\delta, \xi|y))$

That is, for each value of $\delta$, this is the maximum of the likelihood function $L(\theta)$ when $\theta_j$ is fixed at $\delta$.

In most examples I have seen, e.g. Figure 1 from here, reproduced below, the profile likelihood is depicted as a concave function, with a global maximum.

enter image description here

Does this hold in general, even when $\delta$ is a vector of parameters?


1 Answer 1


As a profile likelihood is a very general concept, there can be no reason why this should be true in general. Maybe something can be proved within limited model families?

For a non-convex example on this site, see the answer at https://stats.stackexchange.com/a/510726/11887


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.