Comment on "Including systematic uncertainties in confidence interval construction for Poisson statistics"

The incorporation of systematic uncertainties into confidence interval calculations has been addressed recently in a paper by Conrad [Phys. Rev. D 67, 012002 (2003)]. In their work, systematic uncertainties in detector efficiencies and background flux predictions were incorporated following the hybrid frequentist-Bayesian prescription of Cousins and Highland, but using the likelihood ratio ordering of Feldman and Cousins in order to produce "unified" confidence intervals. In general, the resulting intervals behaved as one would intuitively expect, i.e., increased with increasing uncertainties. However, it was noted that for numbers of observed events less than or of the order of the expected background, the intervals could sometimes behave in a completely counterintuitive fashion-being seen to initially decrease in the face of increasing uncertainties, but only for the case of increasing signal efficiency uncertainty. In this Comment, we show that the problematic behavior is due to integration over the signal efficiency uncertainty while maximizing the best fit alternative hypothesis likelihood. If the alternative hypothesis likelihood is determined by unconditionally maximizing with respect to both the unknown signal and signal efficiency uncertainty, the limits display the expected intuitive behavior.