How to find Influence function? 
Derive $IF(x;T,F)$ when 
  $$\displaystyle T(F)=\int_{F^{-1}(\alpha)}^{F^{-1}(1-\alpha)}x ~dF(x)$$
  Here $IF$ stands for Influence function.

Trial: Here $$\begin{align}IF(x;T,F) &=\lim_{t\to 0}\frac{T((1-t)F+t\Delta_x)-T(F)}t \\ &=\lim_{t\to 0}\frac{g(t)-g(0)}t=\frac{d}{dt}g(t)|_{t=0} \end{align}$$ Then I try to simplify $T(F)$ as $$\int_{F^{-1}(\alpha)}^{F^{-1}(1-\alpha)}x ~dF(x) \\ =\int_{\alpha}^{1-\alpha}F^{-1}(y) ~dy$$ 
Then I am stuck. Please help. 
 A: Let $\mathcal{F}$ denote the set of (absolutely continuous) c.d.f. and let $X$ denote a random variable with c.d.f. $F \in \mathcal{F}$ and support $\mathscr{X} \subseteq \mathbb{R}$. We first give some definitions that follow from the paper of Cowell & Viktoria-Feser (2002): WELFARE RANKINGS IN THE PRESENCE OF CONTAMINATED DATA, Econometrica 70, pp. 1221-1233; hereafter called C&V. This provides us with building blocks to answer the question.


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*Quantile functional: For any $q \in (0,1]$, the $q$th quantile is the functional $Q:\mathcal{F} \times (0,1] \rightarrow \mathscr{X}$, defined by $Q(F,q):= \inf \{x \vert F(x) \geq q \}$. The IF of $Q$ at $x_0 \in \mathscr{X}$ is given in the 1981 book of Peter Huber,


$$IF \big( x_0,Q(\cdot,q),F \big) =\frac{q-\boldsymbol{1}\{Q(F,q) \geq x_0 \}}{f(Q(F,q))},$$
where $f$ denotes the p.d.f. of $F$, and $\boldsymbol{1}\{\ldots\}$ is the indicator function.


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*Cumulative income functional: The $q$th cumulative income functional is the functional $C:\mathcal{F} \times (0,1] \rightarrow \mathscr{X}$ given by
$$C(F,q):=\int^{Q(F,q)} x \mathrm{d} F(x).$$
The IF of $C$ is given by (see C&V, p. 1226)


$$IF \big( x_0,C(\cdot,q),F \big)=q Q(F,q) - C(F,q) + \boldsymbol{1}\{ q \geq F(x_0) \} \big(x_0 - Q(F,q) \big)$$
With this, we have all elements required to compute the IF of your $T(F)$. Note that we have (by the linearity of the integral)
$$T(F) \equiv C(F,1-\alpha) - C(F,\alpha).$$
Hence, the IF of $T(F)$ obtains by "patching things together". 
Comment: the functionals introduced above got their name from analysis of income data.
