MULTIPLE COMPARISONS WITH THE BEST: BAYESIAN PRECISION MEASURES OF EFFICIENCY RANKINGS

(prob. = 0.383). Similarly, we find that G2 rarely dominates G3, suggesting that these
groups are not clearly differentiated at any meaningful level of statistical precision.

Moving on to firm-specific comparisons, the most and least efficient firms can be com-
pared to the remainder of their respective groups to examine whether they are clearly
identified as best and worst. The posterior probability that F1 is less efficient than the re-
mainder of G4 is 0.641, reflecting a reasonable confidence in this ranking, but not definitive
support. The posterior probability that F31 is more efficient than the remainder of G1 is
0.403, indicating that it does not necessarily deserve to be overly singled out as superior,
although this probability still greatly exceeds the expected probability if all firms in G1
were equally efficient (recall that the group contains a total of seven firms).

Finally, proceeding to the comparisons of the index firms to the other groups, we begin
with F1 again. F1 has a posterior probability of being less efficient than G3 equal to 0.933,
of being less efficient than G2 equal to 0.990, and of being less efficient than G1 equal to
0.994. All of these indicate enormous evidence in favor of the precision of the last-place
ranking of this firm relative to all firms in the top 3 groupings. F31 has an estimated
posterior probability of being more efficient than G4 of 0.996, than G3 of 0.917, and than
G2 of 0.834. Thus, F31 appears to be properly ranked above the firms in the other groups.

The BMC comparisons in this section clearly do not exhaust all possible subsets. Our
intent is only to convey the richness of the possible types of comparisons that can be easily
performed.

4.4.2 Comparing Bayesian to Sampling Theory MCB

The basic sampling theory MCB algorithm of Horrace and Schmidt (2000) provides
a set of firms which cannot be statistically differentiated from the uncertain best firm,

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