Let \(F\) be a symmetric and continuous distribution function and consider the functional parameter \(\theta\) that corresponds to the expectation

\[(1-2 \alpha)^{-1} E\left(X \delta\left\{X ;\left[\xi_{\alpha}, \xi_{1-\alpha}ight]ight\}ight),\]

where \(\alpha \in\left[0, \frac{1}{2}ight)\) is a fixed constant, \(\xi_{\alpha}=F^{-1}(\alpha)\), and \(\xi_{1-\alpha}=F^{-1}(1-\alpha)\).

a. Prove that this functional parameter can be written as

\[T(F)=(1-2 \alpha)^{-1} \int_{\alpha}^{1-\alpha} \xi_{u} d u\]

b. Let \(G\) denote a degenerate distribution at a real constant \(\delta\). Prove that

\[\Delta_{1} T(F ; G-F)=(1-2 \alpha)^{-1} \int_{\alpha}^{1-\alpha} \frac{u-\delta\{F(\delta) ;(-\infty, u]\}}{f\left(\xi_{u}ight)} d u\]

c. Prove that the differential given above can be written as

\[\Delta_{1} T(F ; G-F)= \begin{cases}(1-2 \alpha)^{-1}\left[F^{-1}(\alpha)-\xi_{1 / 2}ight] & \delta \in\left(-\infty, \xi_{\alpha}ight) \\ (1-2 \alpha)^{-1}\left(\delta-\xi_{1 / 2}ight) & \delta \in\left[\xi_{\alpha}, \xi_{1-\alpha}ight] \\ (1-2 \alpha)^{-1}\left[F^{-1}(\alpha)-\xi_{1 / 2}ight] & \delta \in\left(\xi_{1-\alpha}, \inftyight)\end{cases}\]