00374 $5c25c1d87a2206bfa2745a636b5976

378

Obenchain

because E[(X - p)^] = 0 and E[(X - p)4]= 3 o⁴ in this special case, Johnson and Kotz (1970), page 47.

The above calculations illustrate the important point that both ER and VR usually increase whenever the process mean of X is off-target. In other words, regrets can be sensitive to much morę than simply the variability in a X characteristic.

The Argument That Choice of Regret Scaline Is Unimnortant Suppose that two regret function parametrizations dififer onJy by a constant, positive, multiplicative "rescaling factor," f, of the form:

R₂(X) = f • Ri(X)    [11]

These two regret functions will be equivalent in the sense that they both will yield the same "unitless" index value. This point is established by observing that

I₂(X)

R₂(X) fR,(X) ER₂ f-ER,

I,(X)

[12]

because the numerator and denominator f-factors cancel each other out. Notę that ER₂ should always be estimated to be f times ERi because the observed R₂(X) and Ri(X) data will always have the exact same pattems of outliers and rational subsets. On the other hand, some very smali numerical differences might be introduced by rounding both ERi and ER₂ to only a very few decimal places.

Ouality Standard Setting Summary In summaiy, the generał "shape" of the regret function may be important, as may be one's choice of scaling along the X axis. But choice of regret units (dollars, yen, marks, francs, etc.) becomes meaningless once we re-express results on our index scalę. Because asymmetric regret functions can be thought of as using a different "scalę" on either side of the target, the specific form of asymmetry can also have a profound effect on the resulting index...unless all observed X values will always fali on only one side of the target! Unbounded regrets (like ąuadratic and absolute value) are generally not sensitive to X-scaling, but they tend to produce regret indices with a heavy, right-hand taił.