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I was reading a text about linear models and regression and came across the statement:

"To find the maximum likelihood estimates for $\theta$ and $\sigma^2$ the log-likelihood must be concentrated with respect to $\sigma^2$." [1]

How does one "concentrate" a function with respect to a some quantity? I don't understand what operation is being referred to here.

[1] "Linear Models and Regression." Pharmacokinetic-Pharmacodynamic Modeling and Simulation, by Peter L. Bonate, 2nd ed., Springer, 2014, p. 63.

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Concentrating the likelihood in one of the parameters means eliminating it and leaving a reduced likelihood function that can be maximized in terms of the other parameter. Typically this is done by taking one of the partial derivatives, setting it equal to zero. And then using that equation to eliminate the parameter.

Reading the article, I see why this is confusing. If you replace the phrase “After concentrating the log-likelihood” with “In order to concentrate the log-likelihood” it makes sense. It seems like a weird error to make, but what they describe afterwards agrees with a procedure I’ve seen called “concentrating” in other sources.

Edit After reading again now that I have more time, it's not super clear that the change in meaning I suggested above makes the text make sense. It's just a not-so-clear account, I think. See these notes, section 1.4 instead.

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