Ferreri Distributions

Probability Distribution: List of Statistical Distributions > Ferreri distributions

Ferreri distributions (distribuzione di Ferreri) are a system of probability distributions described by Ferreri [1].

These univariate distributions have desirable properties such as support on the entire real real line. They are also continuous and unimodal (single-peaked) with exponential tails that are similar to the normal distribution but with different degrees of kurtosis. This makes them important for applied work, according to Ferreri. For example, many biological phenomena have leptokurtic distributions while manufacturing applications require platykurtic distributions. Despite these benefits, they are seldom used for applications, possibly because their shape parameters are challenging to estimate [2].

Ferreri distributions properties

The distributions have probability density functions (PDFs) is a four-parameter ((a, b, c, ξ) distribution of the form [1]:

where
(a, b, c, ξ) are the four parameters: ξ is the mean.

• If c = 1, β₂ < 3;
• If  c = – 1, β₂ > 3.
• As a increases, the distribution approaches a normal distribution [3].

The rth absolute moment about ξ is
While the distribution is usually attributed to Ferreri (1964), there is a mention of the “Ferreri distribution” in the 1959 book International Journal of Abstracts: Statistical Theory and Methods Volume 1 [4]

Due to the age of the source material, I was unable to obtain any originals to confirm more details about the distribution or its origins (Note: Brown University [5] has one copy in offsite storage). Outside of entries in various dictionaries – such as the Oxford Dictionary of Statistical Terms [6] –there is a general dearth of entries about the Ferreri distributions in the literature. This appears to be one of those distributions that is destined to vanish into obscurity.

References

1. Ferreri, C. (1964). A new frequency distribution for single variate analysis, Statistica (Bologna), 24, 223-251. (In Italian)
2. [2] OSAMU FUJITA. FLAT-TOPPED PROBABILITY DENSITY FUNCTIONS FOR MIXTURE MODELS
3. Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.
4. JOURNAL International journal of abstracts, statistical theory and method. Edinburgh : Published for the International Statistical Institute by Oliver and Boyd/ Vol. 1, no. 1 (July 1959)-v. 4, no. 4. c1959-c1963
5. JOURNAL Entry https://bruknow.library.brown.edu/discovery/fulldisplay/alma991020951269706966/01BU_INST:BROWN
6. Dodge, Y. et al. (Eds.). (2006). The Oxford Dictionary of Statistical Terms. Oxford University Press.