§ Expectiles

Mean is a minimiser of $L_2$ norm: it minimizes the loss of penalizing your 'prediction' of (many instances of) a random quantity. You can assume that the instances will be revealed after you have made the prediction. If your prediction is over/larger by $e$ you will be penalized by $e^2$. If your prediction is lower by $e$ then also the penalty is $e^2$. This makes mean symmetric. It punishes overestimates the same way as underestimates. Now, if you were to be punished by absolute value $|e|$ as opposed to $e^2$ then median would be your best prediction. Lets denote the error by $e_+$ if the error is an over-estimate and $e_-$ if its under. Both $e++$ and $e_-$ are non-negative. Now if the penalties were to be $e_+ + a e+-$ that would have led to the different quantiles depending on the values of $a > 0$. Note $a \neq 1$ introduces the asymmetry. If you were to do introduce a similar asymmetric treatment of $e_+^2$ and $e_-^2$ that would have given rise to expectiles.