This is concerning Eq. (3.7) of C R Rao's 1945 paper (see p.81 of this article). Can someone help in figuring out the second equality in Eq. (3.7)?
His claim (since $\phi(x,\theta) = \Phi(T,\theta) \psi(x_1,\dots,x_n)$ from Eq. (3.6)) can be written as $$\theta = \int t \phi \pi dx_i = \int t \Phi(T,\theta) \psi(x_1,\dots,x_n) \pi dx_i = \int f(T)\Phi(T,\theta)dT,$$ for some function $f(T)$ of $T$, independent of $\theta$, where
My question is concerning the last equality. Prof Rao seems to regard $t\psi(x_1,\dots,x_n) \pi dx_i$ as $f(T)dT$. Since $\psi(x_1,\dots,x_n)$ is essentially the conditional distribution of $x_1,\dots, x_n$ given $T$ and since $T$ is a sufficient statistics, it is true that $\psi(x_1,\dots,x_n)$ is a function depending on $T$, but is independent of $\theta$. However, I am not able to see how he obtained the last equality (mainly because of the $t$ which is any unbiased estimate of $\theta$). Any help in this connection is greatly appreciated.