# What is the exact definition of profile likelihood?

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

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–  Alexander May 17 '12 at 15:00

I would suggest

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.

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+1, very clearly explained; I notice you've been doing a lot of that –  gung May 17 '12 at 15:31