Statistics for Process Control Engineers

34Amoroso Distribution

The Amoroso distribution^[28] is defined by the PDF

(34.1)

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

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		1	1
Weibull‐III	α	β	1	k
chi‐squared	0	2		1
gamma	0	β	k	1
Lévy	α			−1
inverse gamma	α	β	δ	−1
Pearson Type III	α	β	δ	1
inverse exponential	α	β	1	−1
stretched exponential ...