  Suppose an observable x is the measured value (negative or non-negative) of a
true mean mu (physically non-negative) in an experiment with a Gaussian
resolution function with known fixed rms deviation s. The most powerful
one-sided upper confidence limit at 95% C.L. is UL = x+1.64s, which I refer to
as the "original diagonal line". Perceived problems in HEP with small or
non-physical upper limits for x<0 historically led, for example, to
substitution of max(0,x) for x, and eventually to abandonment in the Particle
Data Group's Review of Particle Physics of this diagonal line relationship
between UL and x. Recently Cowan, Cranmer, Gross, and Vitells (CCGV) have
advocated a concept of "power constraint" that when applied to this problem
yields variants of diagonal line, including UL = max(-1,x)+1.64s. Thus it is
timely to consider again what is problematic about the original diagonal line,
and whether or not modifications cure these defects. In a 2002 Comment,
statistician Leon Jay Gleser pointed to the literature on recognizable and
relevant subsets. For upper limits given by the original diagonal line, the
sample space for x has recognizable relevant subsets in which the quoted 95%
C.L. is known to be negatively biased (anti-conservative) by a finite amount
for all values of mu. This issue is at the heart of a dispute between Jerzy
Neyman and Sir Ronald Fisher over fifty years ago, the crux of which is the
relevance of pre-data coverage probabilities when making post-data inferences.
The literature describes illuminating connections to Bayesian statistics as
well. Methods such as that advocated by CCGV have 100% unconditional coverage
for certain values of mu and hence formally evade the traditional criteria for
negatively biased relevant subsets; I argue that concerns remain. Comparison
with frequentist intervals advocated by Feldman and Cousins also sheds light on
the issues.
