In this appendix, we derive the inefficiency index E(ui|∈i) of Jondrow et al. (1982), and the technical efficiency index E(exp(−ui)|∈i) of Battese and Coelli (1988). Again, we give detailed derivations assuming u has a truncated-normal distribution. Derivations for models with a half-normal distribution, a truncated normal distribution with scaling property, and an exponential distribution are sketched at the end of this appendix.

For the truncated-normal model, the density function of u|∈ is given by the density of the joint distribution of u and ∈ given by (A.10) over the density of ε given by (A.24). The result is

where μ* and σ* are defined in (A.12) and (A.13), respectively. Then,

Let

Then

Now, for E(exp(−u)|∈),

Note that

therefore, the formula is simplified to

Let

Finally,

To derive the estimates for models with the half-normal distribution, we only need to substitute μ = 0 into the equations. For the truncated-normal model with scaling property, we have shown in (A.26) to (A.28) that the distribution is a reparameterization of a standard truncated-normal distribution. Therefore, we derive the estimates by substituting μ by and σ by, where and are shown in (A.26) to (A.28).

To derive the estimates for an exponential distribution model, we need to know the distribution of u|∈, where u has an exponential distribution. The density, f(u|∈), is obtained by dividing the joint density of u and ε in (A.30) by the density of ∈ in (A.31). After simplifications, the result is

where

Note the similarity between (B.10) and (B.1): The differences are in the definitions of μ* and σ*. Because of this similarity, we can derive the efficiency estimates for the exponential distribution following exactly the same steps outlined in this appendix, except that different definitions of μ* and the σ* are used in the derivation. As a result, the formulas of E(u|∈) and E(exp(−u)|∈) for the exponential distribution model are similar to that of (B.3) and (B.9), respectively, with the only differences being how µ* and σ* are defined.