What is the exact definition of profile likelihood?

Question

Does anyone here know the exact definition of Profile Likelihood? Or does it have one?

Answer 1

I would suggest

Sprott, D. A. (2000). Statistical Inference in Science. Springer. Chapter 4

Next, I am going to summarise the definition of the Profile or maximised likelihood.

Let $\theta$ be a vector parameter that can be decomposed as $\theta = (\delta,\xi)$, where $\delta$ is a vector parameter of interest and $\xi$ is a nuisance vector parameter. This is, you are interested only on some entries of the parameter $\theta$. Then, the likelihood function can be written as

$${\mathcal L}(\theta;y)={\mathcal L}(\delta,\xi;y)=f(y;\delta,\xi),$$

where $f$ is the sampling model. An example of this is the case where $f$ is a normal density, $y$ consist of $n$ independent observations, $\theta=(\mu,\sigma)$ and say that you are interested on $\sigma$ solely, then $\mu$ is a nuisance parameter.

The profile likelihood of the parameter of interest is defined as

$$L_p(\delta)=\sup_{\xi}{\mathcal L}(\delta,\xi;y).$$

Sometimes you are also interested on a normalised version of the profile likelihood which is obtained by dividing this expression by the likelihood evaluated at the maximum likelihood estimator.

$$R_p(\delta)=\dfrac{\sup_{\xi}{\mathcal L}(\delta,\xi;y)}{\sup_{(\delta,\xi)}{\mathcal L}(\delta,\xi;y)}.$$

You can find an example with the normal distribution here.

I hope this helps.