# MLE for mean of symmetric but otherwise unknown distribution

Given i.i.d. draws $x_1,...,x_n$ from $X$, where:

• $X$ has a finite mean $E[X]=\mu$,
• $X$ is symmetric about its mean, meaning $f_X(\mu+c)=f_X(\mu-c)$ for all $c$.,
• The probability density function $f_X$ is not otherwise known.

Is it possible to prove the following?

Proposition. The MLE for the mean of $X$, is the population mean, $\hat \mu_{MLE}=\bar x = \sum_{i=1}^n x_i$.

A proof or a counterexample would be great. I am willing to additionally assume that $X$ has a finite variance $Var[X]=\sigma^2$ if that becomes necessary for the proposition to hold, or if it simplifies the proof.

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Perhaps you can use the invariance property of MLEs in some way? –  hejseb May 13 at 18:26