4. DIFFERENTIABILITY OF FUNCTIONALS T OF F OR P 11 Example 4.4 T(F)=F-1(1/2),and suppose that F has density f which is positive at F(1/2). Then,with F (1-t)F+tG, 品rl=盖'/plo Note that F(F(1/2))=1/2,and hence 0=品('y2 品r'a/2)+G-PG'a2l. =f(F-1(1/2)T(F;G-F)+(G-F)(F-1(1/2)+0, so that T(E;G-F)=-G-FF-11/2 f(F-1(1/2) ∫(1-o.F-1/2(x)-1/2)dG(r f(F-1(1/2) Hence 1 r(o）=ICT,P=-fe-0/2l-,F-1awp1@）-1/2以. Example 4.5 The p-th quantile,T(F)=F-(p).By a calculation similar to that for the median, T(F;G-F)=- (G-F)(F-1(p) f(F-1(p) f(1(-.F-1(p)]()-p)dG(x) f(F-1(p) and 1 r@=1C(工，=f阿l-,F-11@-以. Now we need to consider other types of derivatives:in particular the stronger notions of deriva- tive which we will discuss below are those of Frechet and Hadamard derivatives. Definition 4.3 A functional T:FR is Frechet-differentiable at FEF with respect to d,if there exists a continuous linear functional T(F;)from finite signed measures in R such that (1) IT(G)-T(F)-T(F;G-F)I0 as d.(F,G)-0. d,(G,F) Here are some properties of Frechet-differentiation: Theorem 4.1 Suppose that d,is a metric for weak convergence(i.e.the Levy metric for df's on the line;or the Prohorov or dual-bounded Lipschitz metric for measures on a metric space (S,d)). Then:4. DIFFERENTIABILITY OF FUNCTIONALS T OF F OR P 11 Example 4.4 T(F) = F −1(1/2), and suppose that F has density f which is positive at F −1(1/2). Then, with Ft (1 − t)F + tG, d dtT(Ft) ' ' ' t=0 = d dtF −1 t (1/2) ' ' ' t=0. Note that Ft(F −1 t (1/2)) = 1/2, and hence 0 = d dtFt(F −1 t (1/2)) ' ' ' t=0 = d dt{F(F −1 t (1/2)) + t(G − F)(F −1 t (1/2))} ' ' ' t=0 = f(F −1(1/2))T˙(F; G − F)+(G − F)(F −1(1/2)) + 0, so that T˙(F; G − F) = − (G − F)(F −1(1/2)) f(F −1(1/2)) = − " (1(−∞,F −1(1/2)](x) − 1/2)dG(x) f(F −1(1/2)) . Hence ψF (x) = IC(x; T, F) = − 1 f(F −1(1/2)){1(−∞,F −1(1/2)](x) − 1/2}. Example 4.5 The p−th quantile, T(F) = F −1(p). By a calculation similar to that for the median, T˙(F; G − F) = − (G − F)(F −1(p)) f(F −1(p)) = − " (1(−∞,F −1(p)](x) − p)dG(x) f(F −1(p)) . and ψF (x) = IC(x; T, F) = − 1 f(F −1(p) {1(−∞,F −1(p)](x) − p}. Now we need to consider other types of derivatives: in particular the stronger notions of deriva￾tive which we will discuss below are those of Fr´echet and Hadamard derivatives. Definition 4.3 A functional T : F → R is Fr´echet - differentiable at F ∈ F with respect to d∗ if there exists a continuous linear functional T˙(F; ·) from finite signed measures in R such that |T(G) − T(F) − T˙(F; G − F)| d∗(G, F) (1) → 0 as d∗(F, G) → 0. Here are some properties of Fr´echet - differentiation: Theorem 4.1 Suppose that d∗ is a metric for weak convergence (i.e. the L´evy metric for df’s on the line; or the Prohorov or dual-bounded Lipschitz metric for measures on a metric space (S, d)). Then: