Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 223 / 3 / d / ii / Solution

Assume $d<c$. Directly comparing the two piecewise densities givesSincewe obtainThus the sample log-likelihood ratio is $n{S}_n$ with truncation values $a=\log d$ and $b=\log c$. Rejecting for large $S_n$ is exactly the likelihood-ratio test between the two least-favorable contaminated distributions.