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

set of firms whose estimated efficiency measures are not statistically significantly below
the measure of the uncertain most efficient firm. Those firms for which the condition
in equation (2.2) does not hold are obviously outside that set and can be said to be
statistically less efficient than the best firm.

To extend MCC and MCB to the case in which the estimated TEs are correlated (the
common case), the h described here becomes a function of the firms being compared and so
is replaced by a comparison-specific hj_i . This comparison-specific confidence interval width
is computed by multiplying a firm-specific adjusted critical value dj and a comparison-
specific covariance σ_j∙_i that replaces 2kσ2 in the formula for h from section 2.2.

3. A Bayesian Approach to Measuring the Precision of Efficiency Rankings

In contrast to the sampling theory approach outlined above, we show in this section
that a Bayesian approach can be taken using the empirical results that arise naturally from
the Markov Chain Monte Carlo (MCMC) algorithm employed to derive the numerical es-
timates in our application and from any other numerical Bayesian estimation technique.
This Bayesian Multiple Comparison (BMC) methodology provides exact (posterior) prob-
ability levels for each comparison statement to be evaluated. Thus, rather than simply
stating that “firm A is (not) significantly more efficient than all firms in group 4 using
a 5 percent significance level,” we can make statements along the lines of “there is an
estimated 97 percent probability that firm A is more efficient than all firms in group 4”
and “there is only a 15 percent probability that firm A is more efficient than all firms in
group 4.” These statements contain much more information and a much higher degree of
specificity than the ones developed using the MCC/MCB framework.

The BMC is simple to implement with the parameter draws, generated in our appli-
cation by a Gibbs sampling algorithm, which we use to compute posterior estimates of the

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