Optimal departure time choices as quantiles and expectiles of the travel time distribution

We consider two alternative preference specifications for optimal departure time choices: the classical asymmetric linear loss for early and late delays, and a new specification based on asymmetric quadratic loss. We demonstrate that the resulting optimal choices correspond to the tail indices of the travel time distribution—quantiles for the asymmetric linear specification and expectiles for the asymmetric quadratic specification. Additionally, we establish a correspondence between the choices induced by these two utility specifications, showing that the asymmetric quadratic preference class is a valid alternative with favorable properties, as demonstrated through examples and applications. For both utility specifications, we derive travel time reliability ratios and present a straightforward computation method using the concepts of $\tau$-deviation and $\tau$-variance. Moreover, we test both specifications using departure time choice data, finding that the quadratic loss specification more accurately represents actual departure time choices. Finally, using Montreal travel time data for two alternative routes, we fit a mixture of gamma distributions and compare the behavior of the optimal departure time choices and travel time reliability ratio curves induced by the different utility specifications.