Uncovering a generalised gamma distribution: from shape to interpretation

In this paper, we introduce the flexible interpretable gamma (FIG) distribution which has been derived by Weibullisation of the body-tail generalised normal distribution. The parameters of the FIG have been verified graphically and mathematically as having interpretable roles in controlling the left-tail, body, and right-tail shape. The generalised gamma (GG) distribution has become a staple model for positive data in statistics due to its interpretable parameters and tractable equations. Although there are many generalised forms of the GG which can provide better fit to data, none of them extend the GG so that the parameters are interpretable. Additionally, we present some mathematical characteristics and prove the identifiability of the FIG parameters. Finally, we apply the FIG model to hand grip strength and insurance loss data to assess its flexibility relative to existing models.


Introduction
The best known form of the generalised gamma (GG) distribution was defined by Stacy in 1962 Stacy andHoshkin (1962).Before this, a precursor model had been analysed by Amoroso in 1925 for income distribution modelling purposes Amoroso (1925).The GG contains numerous sub-models, in-cluding the exponential, gamma, Weibull, and log-normal, as limiting cases.The probability PDF of the standard (unscaled) GG is given below: where z, d, p > 0, Γ(•) is the gamma function, and is denoted as Z ∼ GG(p, d).Note that (1) is equivalent to having a = 1 in the PDF of Stacy and Hoshkin (1962).This is done as a simplification, with the knowledge that a simple scaling can be applied after generalisation.The role of the GG distribution shape parameters become apparent when considering the derivative of the log of the kernel in (1), The left tail behaviour for GG is determined by d.Considering the case where, d ̸ = 1 and lim z→0 + d(z; p, d) in ( 2), the first term dominates as the second approaches zero making d the primary shape determinant.If d = 1, it has no affect on the left tail shape.Figure 1 illustrates these GG PDF properties in relation to d.The right tail behaviour for the GG is influenced by p. Considering the case where, lim z→∞ d(z; p, d) in ( 2), the first term approaches zero while the second dominates, thus making p the primary shape determinant.Figure 2 illustrates these GG PDF properties in relation to p.The broad range of distribution shapes for the GG can model has led to widespread application, such as in survival analysis, Lee and Gross (1991), time series Barkauskas et al. (2009), phonemic segmentation Almpanidis and Kotropoulos (2008), wireless fading models Chen et al. (2011), drought Ali et al. (2008), statistical size Kleiber and Kotz (2003), demographic research Kaneko (2003), and economics Amoroso (1925).Building on the success of the GG many authors have improved the applicability of the GG by the addition of parameters through generalisation.Generally speaking, the main excitement of these generalisations is their focus on superior fit in niche applications.In Table 1 a timeline of GG generalisations is given for completeness.
The demands of new models today have a wider focus than simply better fit.The following authors Ley (2015); Jones (2015); McLeish (1982); Punzo and Bagnato (2021); Wagener et al. (2021); Ley et al. (2021) specify these sometimes overlooked desirable qualities for new generalisations: • A low number of interpretable parameters: These include parameters that specifically control distribution shape qualities such as or similar to location, scale, skewness, and kurtosis.• Favourable estimation properties: It is important that the parameters can be estimated properly to ensure correct predictions and inferences from the model.Inferentially speaking, a generalised model for use in diagnosing distribution departure from a common baseline distribution.
• Simple mathematical tractability: Closed-form expressions and simple formulae aid in implementation, computational speed, and providing
• Finite moments: Most real-world measurements require this property.
Upon review of the literature in Table 1, none of the generalisations except one, the κ-GG, have got parameters that are easily interpreted.This is due to these parameters having overlapping influence in distribution shape, which then obfuscates their role in achieving a certain fit.The κ-GG is such a distribution because it has an extra right-tail parameter which give geometric instead of exponential tails.Another drawback of these distributions is that they lack simple formulae.This is because of their complex setup of their generating mechanisms.The latter two points are seen as an opportunity for the derivation of a further generalisation of the GG distribution.
Here, we systematically construct a generalisation of the GG distribution that possesses interpretable parameters, favourable estimation, simple formulae, and finite moments because of its generating mechanism and setup.This is done by Weibullising the body-tail generalised normal distribution (BTN) Wagener et al. (2021) in order to have specific parameters for left-tail shape, body shape, and right-tail shape.
The paper is structured as follows.Section 1 illustrates Weibullisation as a means to generate positive real line distributions from symmetric distributions.Section 1 introduces the flexible interpretable gamma distribution (FIG) generated by the Weibullisation of the BTN baseline distribution; The section further provides the derivations of the PDF, cumulative probability function (CDF), moments, moment generating function (MGF).Section 1 the parameters are proven to be identifiable.Section 1 gives background on maximum likelihood (ML) estimation.Section 1 applies the FIG to hand grip strength and insurance loss data.Section 2 summarises the results and key findings.

Origins of gamma-like distributions
In this section we investigate three different situations that give rise to gamma-like distributions.This insight may then be used to a guide an expansion of GG, with the aim of maintaining the shape clear and interpretable roles of the shape parameters.

Weibullisation
The Weibullisation for a given baseline distribution of a random variable Z occurs when considering the random variable Z 1/ν for ν > 0 Pham-Gia and Duong (1989).If the baseline distribution is selected to be symmetrical, the Weibullisation of the random variable |Z| yields a positive distribution which has direct relationship to the shape of the baseline distribution.This process is illustrated for the GG distribution and its PDF in (1).As |Z| is mathematically equivalent to one side or half of the baseline distribution due to symmetry, our analysis will continue with the latter.The GG distribution emerges when Z follows a generalised normal distribution (GN); see Subbotin (1923).The PDF of the half-GN is given as: where z, s > 0. Note that the half-GN contains the half-normal, half-Laplace, and uniform distributions for shape parameter values equal to s = 2, s = 1, and s = ∞, respectively Subbotin (1923).Let y = z 1/ν , then the PDF of the random variable Y = Z 1/ν is given by: where y, s > 0. Comparing ( 1) and ( 4) shows that Y follows a GG distribution with parameters d = ν and p = sν.In the kernel of ( 4), the right-tail is determined by e −y sν and the left-tail by y ν−1 , refer to Section 1.A visual representation of this generating process of ( 4) is given in Figure 3. Observe that d affects the left-tail shape, for d = 1 we have no change to the half baseline distribution, for d < 1 the left tail density is increased, and for d > 1 the left-tail density is decreased.The left-tail shape does not influence the right tail, apart from a change of overall scale in the Weibullised distribution.
The behaviour of the right tail is left to be determined by the half baseline distribution kernel from (3) in this instance.

Power weighted distributions
The GG distribution (1) also arises in the power weighted kernel of the integrand of the r-th absolute moments of the GN distribution.In general, if the absolute r-th moments of a distribution Z exist and are finite, a positive distribution Y can be generated from it.Consider the absolute r-th moment of Z: where r > 0. The integrand is a valid kernel for a positive support distribution, since its integral is finite by definition.Therefore, a new PDF can be generated by normalising the integrand function with the actual value of the integral, with the new PDF given by: where y, r > 0 and f (•) is the original PDF of Z.To generate the GG PDF in this manner, Z is taken as the GN distribution.By substituting (3) into (6) the generated PDF of Y is given by: where y, r, s > 0. Subsequently, from (7), we have that Y follows a GG distribution with d = r + 1 and p = s.

Scale mixture model
The FIG distribution also arises through a scale mixture of power-function distribution (PF), which is a special case of the beta distribution Johnson N.L. et al. (1970).Even this kind of mixture can not represent the GG, we still include it in this section, since it gives rise to the FIG distribution, which has gamma-like properties.The PDF of the scaled PF distribution is given below: where 0 < x ≤ u, u > 0, ν > 0, and is denoted as X ∼ P F (u, ν).

Theorem:
Let Z ∼ P F (u, ν) and u ∼ GG(β, α), then the PDF of Z is given by:

Proof
Using the pre-defined random variables Z and U .Noting that Z < U by definition, and employing the indicator variable I(•) we have the following: which concludes the proof.

Conclusion
The selection of a baseline distribution, it's important to consider the existing roles of shape parameters and symmetry.This is crucial since any ambiguous roles pertaining to the baseline distribution parameters will inevitably be transferred to the distributions that are subsequently generated.Hence, it is not recommended to use asymmetric distributions due to the uncertainty of the effects of the asymmetry parameter following generalisation.This section presented three different methods of obtaining a GG type distributions.To derive the FIG distribution, our preference lies with the power weighted and scale mixture origins' parameterisations, details of which will be extensively discussed in the subsequent section.

The flexible interpretable gamma distribution
This section consists of the motivation for the chosen FIG baseline distribution and the derivations of the PDF, CDF, moments, and MGF for the standard and scaled FIG distribution.

Baseline distribution
The baseline distribution for the FIG is the BTN distribution.The BTN distribution is a generalisation of the GN and normal distribution which has interpretable parameters, simple mathematical tractability, and finite moments.The latter desirable properties will be transferred to the FIG distribution in the same way the Weibullisation of the GN transferred its properties to the GG.The PDF of the BTN is: where z ∈ R, α, β > 0, and Γ (•, •) is the upper incomplete gamma function; see (Gradshteyn and Ryhzhik (2007), p. 899).The parameters have clear roles, where α determines body shape and β determines the tail shape of the distribution.Note that, for α = β = s, (12) is equivalent to (3), making the GN and its nested models a subset of the BTN; for more details refer to Wagener et al. (2021).Due to the latter fact, the Weibullisation of the BTN will therefore contain the GG distribution for α = β as discussed in Section 1.The absolute moments of the BTN are given by: where r > 0; see Wagener et al. (2021).In Figure 4 the different body shapes for a fixed tail shape can be seen.Similarly, in Figure 5 the different tail shapes for a given body shape is shown.The additional body shape parameter of the BTN specifically enhances the body shape of the GG distribution through Weibullisation in the FIG distribution.Therefore, the additional α parameter has an interpretation and provides information about the body shape of the FIG. Figure 6 illustrates this process of Weibullisation and the effect of the body shape parameter α.Here, the GG and its fixed body shape is shown for α = β = 2. Notice that in the region of the body, 0.5 < z < 1, the shape is determined by α.In the region of the left tail, z < 0.5, the shape is determined by ν.In the region of the right tail, z > 1, the shape is determined by β.Importantly, note that both the tail shapes stay markedly the same for different body shapes α.The FIG has extended the GG body
A depiction of the FIG PDF ( 14) and corresponding baseline PDF is given in Figure 6.The PDF of the scaled FIG, denoted as X ∼ F IG(σ, α, β, ν), is obtained using the transformation X = σZ: where σ > 0. Since the scaled FIG is a generalisation of the GG distribution it has many sub-models which is summarised in Table 2.The CDF of the standard FIG is derived with the definition of a CDF and (14): Applying Lemma 2 of the Appendix to both integrals, we have that: where γ(•, •) is the lower incomplete gamma function; see (Gradshteyn and Ryhzhik (2007), p. 899).Subsequently, the CDF of X ∼ F IG(σ, α, β, ν) is given by the substitution of z = x σ in ( 16).

Mode
The maximum of the standard FIG PDF is given by the maximum of the FIG kernel in ( 14).We consider two cases for obtaining the mode of the FIG.For ν ≤ 1 we have that and which implies that the mode is zero.For ν > 1 we have that Examining the elements inside the brackets in ( 19), we find that for which implies ( 19) is greater than zero for some z.Sine both terms in ( 19) are greater than zero for all z, we investigate the rate of decrease for these function as z increases.The relative rate of decrease for the left and right terms in ( 19) with respect to z is given by which implies that βz α e −z β decrease geometrically faster than (ν−1)Γ α β , z β for z > ln(α)−ln(β) β .Noting that (ν − 1)Γ α β , z β is monotonically decreasing, and βz α e −z β increasing and then decreasing but at a slower relative rate for all z we have that ( 19) is negative from some z onward.Therefore, we have that PDF of the is first increasing and then decreasing for ν > 1 implying a mode which can be numerically calculated by setting (19) to zero.

Tail behaviour
To examine the behaviour of the left and right tail of the FIG in comparison to the GG, we examine the derivative of the difference of log-kernel functions between the FIG and GG as follows: Comparing the left tail behaviour of the FIG to the GG, we evaluate lim z→0 + d(z; α, β, ν) = 0 × 1 − 0 = 0, which suggests that the left tail behaviour of the FIG approximates the shape of the GG distribution for small z.This property is visually confirmed by the bottom left sub-figure of Figure 6.Regarding, the right tail behaviour, we first concentrate on the first term in ( 22) for large values of z.
The limit is of the form zero divided by zero, for which we apply L'Hospitals rule: Substituting the result from ( 23) into lim z→∞ d(z; α, β, ν) = 0, we conclude that the FIG right tail behaviour approximates the shape of the GG distribution for large z.Similarly, this property is visually confirmed by the bottom right sub-figure of Figure 6.It is worth noting that the body shape parameter α does not influence calculated left and right tail limits, suggesting that the left and right tail parameters have maintained their roles and interpretation.

Moments
The r-th moment of the standard FIG is derived from ( 14), and Lemma 3 in the Appendix: Subsequently, the rth moment of X ∼ F IG(σ, α, β, ν) is given by ( 24), and the identity E(X r ) = σ r E(Z r ).

Moment generating function
From the definition of a MGF, ( 14) and the series expansion of the incomplete gamma function (Gradshteyn and Ryhzhik (2007), p. 901).The MGF of the standard FIG is derived by: Subsequently, the MGF of X ∼ BT N (µ, σ, α, β) is given by ( 25), and the identity M X (t) = e tµ M Z (tσ).

FIG identifiability
The FIG distribution is derived with the intent of gaining insights from the fitted parameters to data.It is, therefore, important that the parameters of the FIG PDF (15) are mathematically identifiable.

Theorem:
Let

Proof
The proof follows by method of contradiction, assume the parameters of the FIN are not identifiable.That is, there exist parameters where From the hypothesis (26) and substitution of (15) we have that , ∀x > 0, from which we obtain Assuming for contradiction that ν 1 ̸ = ν 2 .It would now be possible that which is in contradiction with the equality in (26) where Therefore, it must necessarily be that ν 1 = ν 2 .We therefore substitute ν = ν 1 = ν 2 from here on forward.From ( 26), it follows that: where k = Taking the derivative of both sides of (28) with respect to x, we obtain After rearranging, it follows that Assuming for contradiction that α 1 ̸ = α 2 .It would now be possible that which is in contradiction with the equality in (29), since kβ 2 σ 1 α 1 β 1 σ 2 α 2 ∈ R + .Therefore, it must necessarily be that α 1 = α 2 .We therefore substitute α = α 1 = α 2 .Consequently, (29) simplifies to: Assuming for contradiction that β 1 ̸ = β 2 .It would now be possible that which is in contradiction with the equality in (30), since kβ 2 σ 1 α β 1 σ 2 α ∈ R + .Therefore, it must necessarily be that β 1 = β 2 .We therefore substitute β = β 1 = β 2 from here on forward.Consequently, (30) simplifies to: Assuming for contradiction that σ 1 ̸ = σ 2 .It would now be possible that which is in contradiction with the equality in (31), since Therefore, it must necessarily be that σ 1 = σ 2 .In summary, it has been proven that

FIG maximum likelihood equations
The log-likelihood (LL) for a random sample where z i = x i /σ.The derivatives of the individual terms in (32) with respect to the FIG parameters are given by: where is the digamma function, and G is the Meijer's G function; see (Gradshteyn and Ryhzhik (2007), p. 850,902).

FIG applications
In this section the FIG is applied to commonly available benchmark data to compare the flexibility with competing distributions.The competitor models are the inverse Guassian (IG) Samuelson et al. (2006) and generalised inverse Guassian (GIG) Jorgensen (2012).These models are some of the most famous for modelling positive data; see Tweedie (1957) and review paper Folks and Chhikara (1978).For a detailed list where the IG has been successfully implemented; see Seshadri (2012) andJohnson N.L. et al. (1970).The evaluation of fit is done by computing both in-sample and out-of-sample validation metrics.The in-sample statistics are Akaike information criterion (AIC is ) and Bayesian information criterion (BIC is ) computed on the subset of data used for estimation; see Akaike (1974); Schwarz (1978).The outof-sample LL (LL os ) is computed on a 10% subset of data excluded from estimation.This is done to ensure robust goodness of fit analysis and the prevention of over fit of the final models.The application is implemented using packages Rigby and Stasinopoulos (2005), NumPy Harris et al. (2020), Scipy Virtanen et al. (2020), and mpmath Johansson (2018) in R and Python.

Hand grip strength
The data consists of the hand grip strength English school boys available in the gammlss.datapackage, available online at https://cran.r-project.org/web/packages/gamlss.data, accessed 23 July 2022.The summary statistics of hand grip strength are given in   terion and out-of-sample LL os are tabulated in Table 4.In this application, the in-sample metrics are the lowest for the IG and GIG.However, the LL os favours the GG and FIG because they are higher than the IG and GIG.It can therefore be concluded that the distributions perform similarly, with preference to be given to the simpler GG and IG due to parsimony.The fitting of the generalised models as competitors remain important, since we would not know whether a more complex model is necessary if we do not fit one.

Conclusions
In this paper, we address the need for more flexible distributions without compromising on desirable distribution traits for positive data (Section 1).The method of Weibullisation is demonstrated (Section 1) and applied to the BTN distribution to yield the FIG distribution.The FIG has the desirable properties such as a low number of interpretable parameters, simple tractability, and finite moments.We provide many common statistical properties for using the FIG in practice.These are PDF, CDF, moments, MGF, and maximum likelihood estimation equations (Section 1 and 1).Regarding identifiability, a proof that the FIG parameters are identifiable, is provided (Section 1).The applicability of the FIG is demonstrated on hand grip strength and insurance loss data where the FIG provides competitive fit in comparison to the IG and GIG distributions (Section 1).Points for further research may include finite mixture modelling, kernel density smoothing, outlier detection, gamma regression, and reliability modelling.

Figure 1 :Figure 2 :
Figure 1: The GG PDF for different values of left-tail shape parameter d and fixed righttail shape p = 1.

Figure 3 :
Figure 3: Examples of different Weibullisations of the half-GN equivalent to GG with different left-tail shapes d and body shapes p.

Figure 4 :Figure 5 :
Figure 4: Examples of BTN PDFs for different values of body shape α and fixed tail shape β = 2.

Figure 6 :
Figure 6: The baseline BTN and generated FIG PDFs for different body and left-tail shape parameters with fixed right-tail shape are shown in the first row (half baseline distributions) and second row (corresponding generated distributions).
Financial disclosure This work is based on the research supported in part by the National Research Foundation of South Africa Ref.: SRUG2204203965; RA171022270376, UID 119109; RA211204653274, Grant NO. 151035, as well as the Centre of Excellence in Mathematical and Statistical Sciences at the University of the Witwatersrand.The work of the author Muhammad Arashi, supported by the Iran National Science Foundation (INSF) Grant NO.4015320.Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the funders of this work.Lemma 3Let α, β > 0. Then the following integral identity holds true

Table 1 :
A timeline of the GG, its generalisations, and their number of parameters (including a scaling parameter).

Table 3 .
The in-sample cri-

Table 3 :
Summary statistics for hand grip strength data.

Table 4 :
In-and out-of-sample metrics of distributions fitted to hand grip strength data.