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



unknown parameters and the technical efficiency scores. As will be detailed precisely in
the next section, numerical Bayesian techniques rely on random draws from throughout
the parameter space to generate approximate values for parameters of interest, functions
of the model parameters (such as efficiency measures), precision measures, and probability
levels in support of hypotheses of interest. The preciseness of the numerical approxima-
tion is controlled by the choice of the Bayesian numerical technique and the number of
parameter draws generated, so researchers can obtain any desired level of precision.

Reserving the discussion of exactly how to get a set of such draws for the next section,
for now it suffices to establish that given a set of random draws from the posterior density
function of a vector of parameters θ one can estimate the posterior mean of a function of
interest, say g(θ), by the arithmetic mean of the draws. See for example, Tierney (1994).
The technical efficiencies which researchers want to compare are just such a function of
interest and can be expressed as a function of the randomly drawn parameter vector.

Each Gibbs draw is used to compute TE scores for each firm, denoted by T Ei(b) for
firm i and draw b. In addition to using these draws to find posterior means, medians,
standard deviations, they can be compared across firms. To estimate the probability that
firm i is more efficient than firm j, we count the number of draws for which firm i’s TE
score is greater. Formally, for a set of B draws on the TE scores,

B

prob(TEi > TEj) = B-1 H(TEi(b) > TEjb)),               (3.1)

b=1

where H is a logical operator equal to one when the argument is true and zero otherwise.
If one uses a different numerical Bayesian approach that yields draws where weights are
needed to arrive at accurate posterior means (such as in importance sampling), then the
weights would scale the right-hand side of equation (3.1) above. To estimate probabilities



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