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

where h = d(2kσ²)¹/², d is the critical value for the joint two-sided confidence interval
which has been adjusted to account for the multiple comparisons, and k is the factor of
proportionality which scales the identity covariance matrix of θ. For details see Horrace
and Schmidt (2000), equation (6). Tables of critical values for MCC can be found in Hahn
and Hendrickson (1971), inter alia. Horrace and Schmidt (2000) also discuss how to extend
the results to cases where the efficiency estimates are correlated (the most common case).

Given the joint confidence interval, Horrace and Schmidt (2000) identify all firms
that are statistically less efficient than the best firm, along with all the firms that cannot
be differentiated from the best. These two groups of firms are simply those for whom
the joint confidence intervals, respectively, do not and do include a zero difference at the
chosen significance level.

2.2 The MCB Method

The extension from the MCC to the MCB method is that now the best firm is con-
sidered unknown, implying that each firm’s efficiency needs to be compared to a best firm
whose identity is uncertain. Thus in equation (2.1), we would need to replace the fixed
index firm θ_N with an unknown best index firm, θ(_N), in Horrace and Schmidt’s nota-
tion. This somewhat complicates the construction of the joint confidence interval, but
the simplified version of the results is that the set of firms which cannot be statistically
differentiated from the uncertain best firm are those for which

ʌ               ʌ _            . . .                                                                                                                     , .

θj - θi ≤ h, ∀ j = i,                                  (2.2)

where h is the same as in equation (2.1) and represents one half of the width of the
confidence interval. All firms for which the condition in equation (2.2) holds are in the set
of possible best firms as defined by Horrace and Schmidt, although technically this is the

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