15 Statistical Issues in Particle Physics
663
• Short and empty intervals: These can be obtained for certain values of the
observable, without resulting in undercoverage. They are generally regarded as
undesirable for the reasons explained above.
It is not obvious how to incorporate the above desiderata on interval length into an
algorithm that would be useful for choosing among different methods for setting
limits. For different experiments studying the same phenomena (e.g. Dark Matter
searches, neutrino oscillation experiments, etc.) it is worthwhile to use the same
technique for calculating allowed parameter ranges.
15.4.2 Two-Sided Intervals
An alternative to giving upper limits is to quote two-sided intervals. For example, a
68% confidence interval for the mass of the top quark might be 172.6–173.4 GeV/c 2 ,
as opposed to its 95% upper limit being 173.6 GeV/c 2 . Most of the difficulties
and ambiguities mentioned above apply in this case too, together with some extra
possibilities. Thus, while it is clear which of two possible upper limits is tighter, this
is not necessarily so for two-sided intervals, where which is shorter may be metric
dependent; the first of two intervals for a particle’s lifetime τ may be shorter, but the
second may be shorter when the ranges are quoted for its decay rate (= 1/τ ). There
is also scope for choice of ordering rule for the frequentist Neyman construction, or
for choosing the interval from the Bayesian posterior probability density. 14
15.4.3 Feldman-Cousins Approach
Feldman and Cousins’ fully frequentist approach[54] exploits the freedom available
in the Neyman construction of how to choose an interval in the data that contains
a given fraction α of the probability, by using their ‘ordering rule’. This is based
on the likelihood ratio L(x, μ)/L(x, μ best ), where μ best is the physically-allowed
value of μ which gives the largest value of L for that particular x. For values of μ
far from a physical boundary, this makes little difference from the standard central
Neyman construction, but near a boundary the region is altered in such a way as to
make it unlikely that there will be zero-length or empty intervals for the parameter
μ; these can occur in the standard Neyman construction (see Fig. 15.4).
14 A Bayesian statistician would be happy with the posterior as the final result. Particle physicists
like to quote an interval as a convenient summary. For a parameter that cannot be negative and
for which the exclusion of zero is interesting (e.g. testing whether the production rate of some
hypothesised particle is non-zero), an upper limit would always include zero, a lower limit or a
central interval would exclude it and a maximum probability density one would not be invariant
with respect to changes in the functional form of the parameter.
663
• Short and empty intervals: These can be obtained for certain values of the
observable, without resulting in undercoverage. They are generally regarded as
undesirable for the reasons explained above.
It is not obvious how to incorporate the above desiderata on interval length into an
algorithm that would be useful for choosing among different methods for setting
limits. For different experiments studying the same phenomena (e.g. Dark Matter
searches, neutrino oscillation experiments, etc.) it is worthwhile to use the same
technique for calculating allowed parameter ranges.
15.4.2 Two-Sided Intervals
An alternative to giving upper limits is to quote two-sided intervals. For example, a
68% confidence interval for the mass of the top quark might be 172.6–173.4 GeV/c 2 ,
as opposed to its 95% upper limit being 173.6 GeV/c 2 . Most of the difficulties
and ambiguities mentioned above apply in this case too, together with some extra
possibilities. Thus, while it is clear which of two possible upper limits is tighter, this
is not necessarily so for two-sided intervals, where which is shorter may be metric
dependent; the first of two intervals for a particle’s lifetime τ may be shorter, but the
second may be shorter when the ranges are quoted for its decay rate (= 1/τ ). There
is also scope for choice of ordering rule for the frequentist Neyman construction, or
for choosing the interval from the Bayesian posterior probability density. 14
15.4.3 Feldman-Cousins Approach
Feldman and Cousins’ fully frequentist approach[54] exploits the freedom available
in the Neyman construction of how to choose an interval in the data that contains
a given fraction α of the probability, by using their ‘ordering rule’. This is based
on the likelihood ratio L(x, μ)/L(x, μ best ), where μ best is the physically-allowed
value of μ which gives the largest value of L for that particular x. For values of μ
far from a physical boundary, this makes little difference from the standard central
Neyman construction, but near a boundary the region is altered in such a way as to
make it unlikely that there will be zero-length or empty intervals for the parameter
μ; these can occur in the standard Neyman construction (see Fig. 15.4).
14 A Bayesian statistician would be happy with the posterior as the final result. Particle physicists
like to quote an interval as a convenient summary. For a parameter that cannot be negative and
for which the exclusion of zero is interesting (e.g. testing whether the production rate of some
hypothesised particle is non-zero), an upper limit would always include zero, a lower limit or a
central interval would exclude it and a maximum probability density one would not be invariant
with respect to changes in the functional form of the parameter.