§ Expectiles

Mean is a minimiser of L2L_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 ee you will be penalized by e2e^2. If your prediction is lower by ee then also the penalty is e2e^2. This makes mean symmetric. It punishes overestimates the same way as underestimates. Now, if you were to be punished by absolute value e|e| as opposed to e2e^2 then median would be your best prediction. Lets denote the error by e+e_+ if the error is an over-estimate and ee_- if its under. Both e++e++ and ee_- are non-negative. Now if the penalties were to be e++ae+e_+ + a e+- that would have led to the different quantiles depending on the values of a>0a > 0. Note a1a \neq 1 introduces the asymmetry. If you were to do introduce a similar asymmetric treatment of e+2e_+^2 and e2e_-^2 that would have given rise to expectiles.