Let $X_i \sim^{iid} f(x), X \in R$, and $\eta_p = F^{-1}(p)$, where 0<p<1.
I have constructed the following partial M-estimators to estimate $\phi = E[max\{X,\eta_p\}]$:
- $\psi_1 = I\{X_i >\eta_p\} - p$
- $\psi_2 = X_i I\{X_i >\eta_p\} + \eta_pI\{X_i \leq\eta_p\} - \phi$.
To find the asymptotic variance of my M-estimator, I need to take the derivative of $\psi$ and find corresponding expectation under the null distribution i.e. $f_o(x)$, $\eta_{p_{0}}$, $\phi_0$.
Let's take a look at one segment of $E[\frac{d\psi_2}{d\eta_p}]$, I want to find: $$E[\frac{d}{d\eta_p}(X_iI\{X_i >\eta_p\})] |_{f_o(x)}$$ $$=\frac{d}{d\eta_p}\{E[(X_iI\{X_i >\eta_p\}]\} |_{f_o(x)}$$ $$=\frac{d}{d\eta_p}\{\int_{\eta_p}^{\infty} x f(x)dx \} |_{f_o(x)}$$
Can you further simplify this?