Comments on the Unified approach to the construction of Classical confidence intervals

The paper comments on properties of the so-called ”Unified approach to the construction of classical confidence intervals”, in which confidence intervals are computed in a Neyman construction using the likelihood ratio as ordering quantity. In particular, two of the main results of a paper by Feldman and Cousins (F&C) are discussed. It is shown that in the case of central intervals the so-called flip-flopping problem, occuring in the specific scenario where the experimenter decides to quote a standard upper limit or a confidence interval depending on the measurement, is due to an expectation which is not justified. The problem can be easily avoided by choosing appropriate confidence levels for the standard upper limits and confidence intervals. In the F&C paper ”upper limit” is defined as the upper edge of a confidence interval, whose lower edge coincides with the physical limit. With this definition of upper limit (F&C limit), in an approach which uses the likelihood ratio as ordering quantity, two-sided confidence intervals automatically change over to ”upper limits” as the signal becomes weaker (Unified approach). In the present paper it is pointed out that this behaviour is not a special property of this approach, because approaches with other ordering principles, like central intervals, symmetric intervals or highest-probability intervals, exhibit the same behaviour. The term ”Unified approach” is therefore equally well justified for these approaches. The Unified approach is presented in the F&C paper as a solution to the flip-flopping problem. This might suggest that the F&C limit is a standard upper limit. Because the F&C limit can be easily misunderstood as standard upper limit, its coverage properties are investigated. It is shown that the coverage of the F&C limit, if it is interpreted as standard upper limit, depends strongly on the parameter μ to be determined, with values around and & (1 + α)/2, where α is the confidence level of the confidence belt. Differences between the F&C limit and a standard upper limit were already pointed out in the F&C paper. In order to exclude any misunderstanding, it is proposed in the present paper to call the F&C limit ”upper edge of the confidence interval”, even if its lower edge coincides with the physical limit.