Generalizing Koenker's distribution

Koenker (1993) discovered an interesting distribution whose α quantile and α expectile coincide for every α   in (0,1)(0,1). We analytically characterize the distribution whose ω(α)ω(α) expectile and α   quantile coincide, where ω(·)ω(·) can be any monotone function. We further apply the general theory to derive generalized Koenker's distributions corresponding to some simple mapping functions. Similar to Koenker's distribution, the generalized Koenker's distributions do not have a finite second moment.