Before revision:   Illustrating the formulation of expectile in terms of expected shortfall, we also provide explicit or semi-explicit expressions of expectile  for some classical distributions.
After revision:  We use concentration inequalities to illustrate how the empirical estimation of tail index using expectile versus expected shortfall is better than expectile versus quantile for fat tail distributions. Illustrating the formulation of expectile in terms of expected shortfall, we also provide explicit or semi-explicit expressions of expectile for some classical distributions.