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Statistics for Process Control Engineers by Myke King

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34Amoroso Distribution

The Amoroso distribution[28] is defined by the PDF

Figure 34.1 shows, with α set to 0 and β to 1, the effect of varying δ1 and δ2.

f(x) vs. x displaying three curves for δ1 = 1, δ2 = 2; δ1 = 2, δ2 = 1; and δ1 = 1, δ2 = 1.

Figure 34.1 Amoroso: Effect of δ1 and δ2 on shape

As mentioned in Section 31.3, the generalised gamma distribution is a special case of the Amoroso distribution in which α is set to zero. This, however, is just a trivial example of what other distributions can be represented. Table 34.1 lists such distributions that are covered elsewhere in this book, showing how each PDF is derived from Equation (34.1).

Table 34.1 Distributions represented by the Amoroso distribution

distribution α β δ1 δ2
shifted exponential α β 1 1
standard exponential 0 images 1 1
Weibull‐III α β 1 k
chi‐squared 0 2 images 1
gamma 0 β k 1
Lévy α images images −1
inverse gamma α β δ −1
Pearson Type III α β δ 1
inverse exponential α β 1 −1
stretched exponential ...

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