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

are statistically tied with each other even if the point estimates of their relative efficiencies
differ.

In this paper we offer a Bayesian alternative that we will argue is simpler to imple-
ment, more flexible over possible comparisons, and provides greater, and more intuitive,
information content. The Bayesian method easily allows comparisons between single firms,
a firm versus a group, or a group versus a group. Further, rather than simply answering
the question of “can we differentiate?” with a yes/no (reject/do not reject), the Bayesian
method provides an estimated probability in support of the rankings ability to differenti-
ate between the two firms or groups compared. Thus, statements such as “firm A can be
ranked as more efficient than firm B with a 92 percent posterior probability” are possi-
ble. We demonstrate this method on technical efficiency rankings of a set of U.S. electric
generating firms derived within a distance function framework.

The remainder of this paper is organized as follows. In section 2, we review the MCB
and MCC approaches pioneered by Horrace and Schmidt for the purposes of efficiency
rankings. In section 3, we introduce the Bayesian approach and discuss differences and
potential advantages to the Bayesian methodology. Section 4 discusses the model, the data,
an overview of the derivation of our efficiency rankings, and the results of our empirical
application. In particular, we focus on the results produced by the Bayesian multiple
comparison approach and contrast them with the original MCC and MCB approaches.
Conclusions follow in section 5.

2. MCB and MCC Approaches to Testing Efficiency Rankings

Horrace and Schmidt (2000) pioneered the use of MCB and MCC in creating statistical
confidence intervals for use with comparisons of multiple firm efficiency scores. Their

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