Quantile methods Class Notes

Asymmetric absolute loss Let us define the “check” function (or asymmetric absolute loss function). For τ ∈ (0, 1) ρτ (u) = [τ1 (u ≥ 0) + (1− τ)1 (u < 0)]× |u| = [τ − 1 (u < 0)]u. Note that ρτ (u) is a continuous piecewise linear function, but nondifferentiable at u = 0. We should think of u as an individual error u = y − r and ρτ (u) as the loss associated with u.1 Using ρτ (u) as a specification of loss, it is well known that qτ minimizes expected loss: s0 (r) ≡ E [ρτ (Y − r)] = τ Z ∞