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Regret Indices and Capability Quantification

of these approximations via probability plotting (using P-P plots for the discrete, Poisson case and Q-Q plots for the continuous, gamma case), as illustrated below. And, once we have confirmed that these approximations are adequate, we can use the "additivity property" of Poisson and standardized gamma distributions to form composite regrets over time and/or across processes, also illustrated below.

Let us now consider two key examples of Poissonization:

EXAMPLE: Readers may wish to verify for themselves the following results for goal-posts regret. If the probability of nonconformance is p, then standard Bemoulli trial calculations yield ER = p and VR = p(l - p).

[The resulting index value for a single observation would thus be either I =

0 or I = 1/p.J The corresponding equivalent nonconformance number is then either EN = 0 or EN = 1/(1 - p), and the equivalent expectancy is EE=p/(l-p). Thus, at least when p is close to zero, goal-posts regret corresponds approximately to EE = p and EN = observed attribute (either 0 or 1.)

EXAMPLE: In ąuadratic loss calculations, equation [9] will simplify to ER = s^ and eąuation [10] will simplify to VR = 2s* when X is normally distributed with mean value on-target, p = T. The regret index for a single

observation would then be I(X) = (X-T)² /o². The resulting equivalent expectancy is EE = 0.5 because equivalent nonconformance would then be EN(X) = (X-T)²/2ct².

Poissonization thus provides an easy-to-explain as well as theoretically sound answer to the question of Gunter (1990), "What is the information content of a measurement?" For example, if you monitor a process using a pass/fail test with your ąuality standard for fraction nonconforming set at p = 0.004 (four-tenths of 1%), then you would need to test 125 units in order to accumulate a pass/fail equivalent expectancy of EE = 0.5. You can get that same amount of information, EE = 0.5, from a single normally distributed measurement by specifying a target value and using ąuadratic regret instead of your pass/fail test!

Smoothing Regret Distributions

Process capability analyses freąuently attempt to "smooth" an observed process distribution by superimposing a fltted parametric distribution on top of