I'm interested in estimating the mean $\mu$ of a distribution given i.i.d samples. The empirical mean is a totally acceptable estimator (and can even be shown to be asymptotically/close to optimal under certains models) for the typical squared loss function.

But now let's say my loss function is very unbalanced/asymmetrical (underestimating is much more costly than overestimating for example), so that the empirical mean is not acceptable anymore. For example, take: $$L(\mu,\hat{\mu})= \frac{2 \ \mu^2 + \hat{\mu}^2}{3 \ \mu^{\frac{4}{3}}\hat{\mu}^{\frac{2}{3}}}$$ which is a positive convex function which is minimized when $\hat{\mu} = \mu$ and is quite unbalanced.

How is this field called? Can anyone point me to a good/seminal paper about this kind of question so I can work my way from there?

Thank you in advance!

PS: It does not have to be about the mean specifically, nor about this loss function.

PS2: I am only interested in a nonparametric setting, so no parametric methods please.