Question on maximum likelihood

[Simfish]

http://en.wikipedia.org/wiki/Maximum_likelihood


What exactly does the "arg" here mean? It seems to be an unnecessary - the max L(\theta) seems to be sufficient enough. Or am I missing something?
$$\widehat{\theta} = \underset{\theta}{\operatorname{arg\ max}}\ \mathcal{L}(\theta).$$

 

[EnumaElish]

arg is needed because the left-hand side is the argument of L.

One can write $$L(\widehat{\theta}) = \underset{\theta}\max\ {L(\theta)}.$$.

Or one can write $$\widehat{\theta} = \underset{\theta}{\operatorname{arg\ max}}\ {L}(\theta).$$

 

[tacman]

The "arg" in this context stands for "argument" and it is used to indicate that the maximum likelihood estimate is the value of the parameter, \theta, that maximizes the likelihood function, \mathcal{L}(\theta). In other words, it is the argument that gives the maximum value of the likelihood function. This notation is commonly used in mathematical and statistical contexts to indicate the input or variable that results in a maximum or minimum value of a function. It is not necessary, but it helps to clarify the purpose of the notation and the meaning of the estimate.