I can calculate a likelihood ratio statistic to measure how well my data fits either of two mutually exclusive models:


$\Lambda({\text{data}}) = \frac{ f( \theta\_0 | {\text{data}} )}{f( \theta\_1 | {\text{data}} )} $


Calculating this statistic is straightforward to me, but I'd like some measure of statistical confidence to be carried along with the likelihood ratio. Is there something analogous to a confidence interval for a likelihood ratio? Or is all of the information about the relative confidence of each model?



Confidence intervals for likelihood ratios
Yes, in principle, the distribution function of the likelihood ratios statistic can be computed, since it is a function of the random variables "data" and the model parameters theta\_0 and theta\_1. Suppose, that cumulative distribution function is calculated, the p-value can be computed giving a confidence estimate.
One example where the distribution of a likelihood ratio is known is when the distribution function "f" is Gaussian (For example in the case of Anova). In this case the likelihood ratio is [F-distributed](https://en.wikipedia.org/wiki/F-test)  (it is a ratio of two chi-squared random variables) and the p-values can be explicitely computed.


In the case where "f" is not Gaussian , in general there is no explicit expression for the distribution function of the likelihood ratio, however, asymptotic distributions in the case of large samples can be obtained, see for example the following article by: [S.S. Wilks](https://projecteuclid.org/journals/annals-of-mathematical-statistics/volume-9/issue-1/The-Large-Sample-Distribution-of-the-Likelihood-Ratio-for-Testing/10.1214/aoms/1177732360.full), where a chi squared asymptotic distribution of the log-likelihood is obtained.


