Likelihood-ratio confidence interval ($2[\ell (\hat{\theta })-\ell (\theta )]\le {\chi }_{1,1-\alpha }^2$)

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 ${\chi }_{1,1-\alpha }^2/2$ below its maximum. This explains the horizontal cutoff on a Box–Cox transformation likelihood plot.