APPENDIX

Chapter X - Maximum Performance Efficiency Approaches for Estimating Best Practice Costs
Data Mining: Opportunities and Challenges
by John Wang (ed)
Idea Group Publishing 2003

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Proof of Theorem 1:

1. Consider the potentially feasible solution m⁰ = k = a⁰_r for all r. Then,

   

   Thus the solution m⁰ = k = a⁰_r is feasible for problem MPE (λ) for all λ > 0, for k⁰ = min (Σy_rj)⁻¹ which is positive due to the positivity of the data. It follows that.

   

2. Here we note that the a_r* (λ) for the MMPE model, while feasible, are not necessarily optimal for the original MPE model. Hence use of these values in the MPE objective function will generally give a lower objective function value.

Proof of Theorem 2:

This is a modification of a proof for a version of the theorem given in Troutt (1993). By the assumption that x is uniformly distributed on {x:w(x) = u}, f(x) must be constant on these contours; so that f(x) = φ(w(x)) for some function, φ( ). Consider the probability P( u ≤ w(x) ≤ u + ε ) for a small positive number, ε. On the one hand, this probability is ε g(u) to a first order approximation. On the other hand, it is also given by

Therefore

Division by ε and passage to the limit as ε → 0 yields the result.

Further Details on the Simulation Experiment

To simulate observations within each data set, a uniform random number was used to choose between the degenerate and continuous portions in the density model

where p =0.048, α =1.07, and β =0.32. With probability p, δ(0) was chosen and w =0 was returned. With probability 1-p, the gamma (α,β) density was chosen and a value, w, was returned using the procedure of Schmeiser and Lal (1980) in the IMSL routine RNGAM. The returned w was converted to an efficiency score, v, according to v =exp{−w^0.5}. For each v, a vector Y was generated on the convex polytope with extreme points e₁ =(v/a₁*,0,0,0), e₂ =(0,v/a₂*,0,0), e₃ =(0,0,v/a₃*,0) and e₄ =(0,0,0,v/a₄*) using the method given in Devroye (1986).

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