Likelihood-ratio confidence interval
= Likelihood-ratio confidence interval
{title2=$2[\ell(\hat\theta)-\ell(\theta)]\leq\chi^2_{1,1-\alpha}$}
Invert a <likelihood-ratio test> for a scalar parameter, maximizing over nuisance parameters for each fixed candidate value. Under regularity, a level $1-\alpha$ interval includes values whose profile <log-likelihood> is at most $\chi^2_{1,1-\alpha}/2$ below its maximum. This explains the horizontal cutoff on a <Box–Cox transformation> likelihood plot.