CDF and PDF of a likelihood ratio random variable?

Question

I am trying to figure out what the CDF and PDF are of a log likelihood ratio test. My research online has led me to believe that they follow a $\chi^2$ distribution according to Wilks:

Given an observed sequence $Y_n$ which is IID according to either hypothesis $H_0$ & $H_1$, where:

$\Lambda = log\frac{f_{Y|H_1}(Y)}{f_{Y|H_0}(Y)} $

is the likelihood ratio. Is the distribution (PDF and CDF) of $\Lambda$ $\chi^2$? How do I find out the degrees of freedom?

Answer 1

1. The distribution is $\chi^2$ only asymptotically and only if $H_0$ is true, finite sample distribution generally depends on distribution of $\{Y_n\}_{n=1}^{N}$.
2. The number of degrees of freedom depends on hypothesis. For simple $H_0: \theta = \theta_0$, it is the number of dimensions in $\theta$, for general $g(\theta) = 0$ it should be like the rank of Jacobian of $g(\theta)$.