What is the exact definition of profile likelihood?

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


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 θ be a vector parameter that can be decomposed as θ=(δ,ξ), where δ is a vector parameter of interest and ξ is a nuisance vector parameter. This is, you are interested only on some entries of the parameter θ. Then, the likelihood function can be written as


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, θ=(μ,σ) and say that you are interested on σ solely, then μ is a nuisance parameter.

The profile likelihood of the parameter of interest is defined as


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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