I am wondering if there is an existing theorem for the distribution of a likelihood ratio test statistic under the alternative. By Wilks theorem, we know the distribution under the null. However, under the alternative, I presume it is noncentral chisquare, but I can't seem to find anything on what the noncentrality parameter is.
I am specifically interested in the setting of Cox PH regression, where the density and hypotheses are as follows:
$$h(t) = h_0(t)exp(\beta X)$$
$$H_0: \beta = 0$$ $$H_1: \beta \neq 0$$
Using the partial likelihood of the Cox model, we can calculate the following likelihood ratio test statistic:
$$\lambda_LR = -2[l(\theta_0)-l(\hat{\theta})]$$
Any help would be much appreciated! Thank you!