Let $X$ be a continuous random variable taking value at $\mathbb R$ with distribution $F_\theta$ and density $f_\theta.$ Define a function $\psi_\theta:\mathbb R\mapsto \mathbb R$ by $$ \psi_\theta(x)=\Phi^{-1}(F_\theta(x)),\quad \forall x\in\mathbb R, $$ where $\Phi$ is the cummulative standard normal distribution function.
I wish to prove $$ E(\psi_\theta(X)\times\frac{d\psi_\theta(X)}{d\theta})=0. $$
Note that $$\frac{d\psi_\theta(X)}{d\theta}=\frac1{\phi(\Phi^{-1}(F_\theta(X)))}\frac{dF_\theta(X)}{d\theta}.$$ But then, how will I compute $\frac{dF_\theta(X)}{d\theta}?$ Any suggestion or helps?
For every $\theta$, the change of variable $t=F_\theta(x)$ yields $$E(\psi_\theta(X)^2)=\int_{-\infty}^\infty\Phi^{-1}(F_\theta(x))^2\mathrm dF_\theta(x)=\int_0^1\Phi^{-1}(t)^2\mathrm dt.$$ The RHS does not depend on $\theta$ (and is equal to $1$) hence, differentiating, one gets $$2E\left(\psi_\theta(X)\frac{\partial\psi_\theta}{\partial\theta}(X)\right)=\frac{\mathrm d}{\mathrm d\theta}E(\psi_\theta(X)^2)=0.$$ The continuity of every CDF $F_\theta$ is necessary but not the existence of PDFs.