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

for multiple comparisons, simply replace one or both of the single TE scores with the sets
desired. For example, to compare firm i to a group J, the logical operator would evaluate
the truth of TE_i^(b) > max{TE⁽_j^b),j ∈ J}.

These probability levels essentially create a Bayesian analog to the MCC procedure,
with the advantage of simplicity and greater information content on the strength of support
in favor (or against) differentiation between compared firms or groups. However, since the
index firm has been fixed, an extension of the above procedure is necessary to generate
a Bayesian MCB. While the frequentist idea of joint confidence intervals for differences
between TE scores does not translate perfectly into the Bayesian framework, one could
create a Bayesian analog. Rather than create a single analog, we choose to list several
possible Bayesian MCB-type measures.

Defining J as the set of all firms other than firm i and retaining the above definition
for the logical operator H , one can estimate the probability that firm i is the most efficient
firm by

B

prob(TEi = TEmax) = B^-1 ∑ H(TE(^b) > max{TEj^b),j ∈ J}).        (3.2)

b=1

Given some value δ , chosen perhaps to represent an economically significant difference
in TE scores, one can compute the probability that a firm’s TE score lies within the
specified range of the best:

B

prob(TE_max - TEi ≤ δ) = B^-1 ∑ H(max{TEj^b) ,j ∈ J} - TE(b) ≤ δ).      (3.3)

b=1

This equation can clearly be used to create an analog to the MCB procedure’s set S
of firm’s in contention to be the best. Simply allow i to vary for firms i = 1, . . . , N and

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