Let $X_1,\ldots,X_n$ be i.i.d. random variables, and $f,g$ two functions. I am wondering whether the following result holds
$$ \mathbb{E}\left[\sum_{i=1}^n \big|f(X_i) - \mathbb{E}[g(X_i)]\big|\right] \leq \mathbb{E}\left[\sum_{i=1}^n\big|f(X_i) - g(X_i)\big|\right]. $$
My derivation is as follows. Start by defining $\epsilon$ as
$$\epsilon = \sum_{i=1}^n \big|f(X_i) - \mathbb{E}[g(X_i)]\big|.$$
Now, it is clear that
$$ \begin{align} \mathbb{E}[\epsilon] &= \sum_{i=1}^n\mathbb{E}\Big[\big|f(X_i) - \mathbb{E}[g(X_i)]\big|\Big] \\ &= \sum_{i=1}^n\mathbb{E}\Big[\big|f(X_i) - g(X_i) + g(X_i) - \mathbb{E}[g(X_i)]\big|\Big]. \end{align} $$
For simplicity, define $\delta_i \equiv f(X_i) - g(X_i) + g(X_i) - \mathbb{E}[g(X_i)]$ and replace above:
$$ \mathbb{E}[\epsilon] = \sum_{i=1}^n\mathbb{E}\big[|\delta_i|\big] = \sum_{i:\:\delta_i \geq 0}\mathbb{E}\big[\delta_i\big] - \sum_{i:\:\delta_i < 0}\mathbb{E}\big[\delta_i\big]. $$
Since $\mathbb{E}\big[\delta_i\big] = \mathbb{E}\big[f(X_i) - g(X_i)\big] \leq \mathbb{E}\Big[\big|f(X_i) - g(X_i)\big|\Big]$, we get
$$ \begin{align} \mathbb{E}[\epsilon] &= \sum_{i:\:\delta_i \geq 0}\mathbb{E}\big[\delta_i\big] - \sum_{i:\:\delta_i < 0}\mathbb{E}\big[\delta_i\big] \\ &\leq \sum_{i:\:\delta_i \geq 0}\mathbb{E}\Big[\big|f(X_i) - g(X_i)\big|\Big] + \sum_{i:\:\delta_i < 0}\mathbb{E}\Big[\big|f(X_i) - g(X_i)\big|\Big] \\ &= \sum_{i=1}^n\mathbb{E}\Big[\big|f(X_i) - g(X_i)\big|\Big] \\ &= \mathbb{E}\left[\sum_{i=1}^n\big|f(X_i) - g(X_i)\big|\right]. \end{align} $$
Putting everything together, we can conclude that
$$ \mathbb{E}[\epsilon] = \mathbb{E}\left[\sum_{i=1}^n \big|f(X_i) - \mathbb{E}[g(X_i)]\big|\right] \leq \mathbb{E}\left[\sum_{i=1}^n\big|f(X_i) - g(X_i)\big|\right]. $$
It sounds like it should not be true, but I am not sure where I might have gone wrong.