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

procedures allow some hypothesis tests to be conducted in a sampling theory framework
so that researchers can state whether a firm is “significantly” more efficient than some
group of firms.

While Horrace and Schmidt (2000) focuses on MCB, MCC seems the more natural
application. Referring to efficiency rankings for concreteness, the distinction is that MCC
involves comparing the estimated efficiency of a chosen (and fixed) firm to another firm or
group of firms while MCB adjusts for the case where the “best” or index firm is unknown.
It is clear that once one recognizes the stochastic and imprecise nature of the estimated
rankings, one should also realize that the most efficient (best) firm is unknown. However,
in most real world application (as opposed to academic ones), it is quite reasonable to use
the firm estimated to be best as the index firm and investigate how many of the other
firms can be declared statistically less efficient. Choosing this index firm as fixed leads one
to the MCC algorithm, so we explain that first.

2.1 The MCC Method

Begin by denoting the estimated measure for each firm i (technical efficiencies in the
application to follow) by θ_i, i = 1, . . . , N. Assume for simplicity that firms were ordered
in such a way that θN has the largest measure (highest efficiency) and is thus the best,
or index, firm against which we wish to compare the others. The MCC method computes
a joint confidence interval of a desired probability level for all the differences between
individual firm efficiencies and the best. That is, for the vector [θN - θ1 , θN - θ2, . . . , θN -
θ_N-1]. When the efficiency estimates are independent, this joint confidence interval can
be given by equation (5) from Horrace and Schmidt (2000), rewritten slightly here as

(1 - α) = Prob(θN - θi - h ≤ Θn - θi ≤ Θn - θi + h, ∀i = 1,N - 1),     (2.1)

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