What Does arg Mean in Maximum Likelihood Estimation?

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SUMMARY

The term "arg" in the context of Maximum Likelihood Estimation (MLE) refers to the argument that maximizes the likelihood function, denoted as \(\widehat{\theta} = \underset{\theta}{\operatorname{arg\ max}}\ \mathcal{L}(\theta)\). This notation is essential as it specifies that \(\widehat{\theta}\) is the value of \(\theta\) that maximizes the likelihood function \(\mathcal{L}(\theta)\). While one could express the result as \(L(\widehat{\theta}) = \underset{\theta}\max\ {L(\theta)}\), using "arg" clarifies the relationship between the parameter and the maximization process.

PREREQUISITES
  • Understanding of Maximum Likelihood Estimation (MLE)
  • Familiarity with likelihood functions
  • Basic knowledge of mathematical notation and optimization
  • Experience with statistical inference concepts
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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).
 
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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).
 

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