On Anomaly Ranking and Excess-Mass Curves, Supplementary Material

Let t > 0. Recall that EM∗(t) = α(t) − tλ(t) where α(t) denote the mass at level t, namely α(t) = P(f(X) ≥ t), and λ(t) denote the volume at level t, i.e. λ(t) = Leb({x, f(x) ≥ t}). For h > 0, let A(h) denote the quantity A(h) = 1/h(α(t + h) − α(t)) and B(h) = 1/h(λ(t + h) − λ(t)). It is straightforward to see that A(h) and B(h) converge when h→ 0, and expressing EM∗ ′ = α′(t)−tλ′(t)−λ(t), it suffices to show that α′(t)−tλ′(t) = 0, namely limh→0A(h)−t B(h) = 0. Now we have A(h) − t B(h) = 1 h ∫ t≤f≤t+h f − t ≤ 1 h ∫ t≤f≤t+h h = Leb(t ≤ f ≤ t+h)→ 0 because f has no flat part.