Amoroso distribution / Wilson-Hilferty distribution

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What is the Amoroso distribution?

The Amoroso distribution (also called the Stacy-Mihram distribution) is an alternate parameterization or “Weibullization” of the generalized gamma distribution. It is a good fit for skewed and heavy-tailed data, but its parameters can be challenging to estimate because different sets of model parameters can give similar fits of the Amoroso distribution [1]. Thus, it is not as well studied as similar distributions such as the exponential distribution (which is itself a special case of the Amoroso).

The Amoroso distribution is named after the Italian mathematician and economist Luigi Amoroso, who first proposed it in 1925 for analyzing and modeling economic income data [2].

A random variable Y follows a four-parameter Amoroso distribution with probability density function (PDF) [1] 

pdf amoroso distribution

where support = { y ≥ µ if a > 0; y ≤ µ if a < 0 } and

Note that the support of the distribution is µ < y < ∞ in Amoroso’s original paper.

Other parameterizations of the distribution exist; Some authors use notation. For example, Crooks describes the PDF as [4]:

pdf amoroso distribution crooks

Note the inclusion of both “a” and “α” as variables in this version, which could confuse depending on how browsers render the letter a (it could resemble the Greek letter alpha (α)).

Other names for the Amoroso distribution

The Amoroso distribution is sometimes called the four-parameter generalized gamma distribution. It is more precisely (albeit wordily!) named the lower-bounded, four-parameter, shifted generalized Gamma distribution [4].

The generalized gamma distribution is one of more than 50 special cases or limits of the Amoroso distribution [3], which include the Weibull distribution, log-normal distribution, gamma distribution, the power law, log-gamma distribution, and the normal distributions [4].

Wilson-Hilferty distribution

The Wilson-Hilferty distribution (also called the WH distribution) is a two-parameter special case of the Amoroso distribution. It is ideal for modelling data with increasing, decreasing and bathtub shape hazard rates. The distribution is named after a statistical technique described by Wilson and Hilferty [5], the so-called Wilson-Hilferty transformation, which gives a normal approximation to the cube root of a chi-squared variable, closely approximating their p-values.

The probability density function (PDF) for the Wilson-Hilferty distribution is [6]

wilson hilferty distribution pdf

Wilson-Hilferty distribution uses

Many authors have expanded on the Wilson-Hilferty transformation including: Ishikawa et al. [7], who used it to improve the accuracy of likelihood analysis where only a small number of modes are available in power spectrum measurements; Zemzami and Benaabidate [8] who considered it to reduce the effect of local variations in neural networks. Other parameterizations of the Wilson-Hilferty distribution include Dey and Elshahhat’s form [9].


[1] Combes, C. & Ng, H. On Parameter Estimation for Amoroso Family of Distributions. Version of Record:

[2] Amoroso, L. (1925). Richerche intorno alla curve die redditi. Ann. Mat. Pura Appl. 21, 123–159.

[3] Hendrik Rogier. Generalized Gamma-Laguerre Polynomial Chaos to Model Random Bending of Wearable Antennas. IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS.

[3] E. W. Stacy, “A generalization of the gamma distribution,” Ann. Math.
Statist., vol. 33, no. 3, pp. 1187–1192, Sep. 1962

[4] Crooks, G. (2010). The Amoroso distribution.

[5] Wilson EB & Hilferty, NM. (1931). The distribution of chi-square. Proc Natl Acad Sci 17(12): 684-688.

[6] Kyurkchiev et al. A note on the Wilson-Hilferty distribution. International Journal of Differential Equations and Applications. Vol 18 No.1 (2019). Retrieved May 23, 2024 from:

[7] Ishikawa, T. et. al. (2014). On the systematic errors of cosmological-scale gravity tests using redshift-space distortion: non-linear effects and the halo bias. Mon Notices Royal Astron Soc 443(4): 3359-3367.

[8] Zemzami, M. and Benaabidate, L. (2016.) Improvement of artificial neural networks to predict daily streamflow in a semi-arid area. Hydrol Sci J 61(10): 1801-1812.

[9] Dey and Elshahhat. (2022). Analysis of Wilson-Hilferty distribution under progressive Type-II censoring. Quality and Reliability Engineering InternationalVolume 38, Issue 7 p. 3771-3796

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