Given that $X_{1},...,X_{n}$ are i.i.d random variables with joint distribution $f(x\mid \theta) $ with 1 dimensional parameter $\theta$ and let $\hat\theta$ be the maximum likelihood estimator of $\theta$.
Based on the Wilks theorem,under null hypothesis that $H_{0}: \theta=\theta_{0}$ ,one have that $-2\log\left(f(x\mid \theta_{0})/f(x\mid \hat\theta)\right)\to \chi_{1}^{2}$ as $n\to\infty$.
Since we know that $E\chi_{1}^{2}=1$, is there anyway I can find the value of the following integral (expectation of log likelihood ratio) or at least asymptotics
$$-2\int f(x\mid \theta_{0})\log\left(\frac{f(x\mid \theta_{0})}{f(x\mid \hat\theta)}\right)\,dx=~?$$
My guess is that the integral should be around 1. Anyone please share some idea or references with the above integral.