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

Appendix: Pseudo Code for Step 1 of the Gibbs Sampler

a. First, generate a Wishart random variable A(i) = [Σ(i) ]^-1 ~ W (S^-1, ν ).

i. Let LL = S^-1, where L is a lower triangular matrix from the Cholesky de-
composition of S^-1. We obtain S^-1 as the inverse of the estimated variance-
covariance matrix of the error terms.

ii. Assume that Q ~ W (I, ν ). Then LQL ~ W (LL, ν ) = W (S^-1, ν )

iii. Now from Anderson (1984) UU ' = Q ~ W (I, ν ), where U is lower triangular,
all u_ij are independent, ui_j∙ ~ N(0,1), i > j, and u²i ~ χ²(ν — i + 1) random
variable, which implies that LUU 'L' ~ W (S^-1, ν ). Thus, we draw uij values
and form Q(i) = U(i)U(i) .

iv. Now use L and Q(i) compute LQ(i)L'∕ν = A(i).

b. Compute Σ⁽ⁱ⁾ = [A⁽ⁱ⁾]^-1 , which is an inverse-Wishart random variable.

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