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Communication

The Gamma-G Family: Brief Survey and COVID-19 Application

by
Gauss M. Cordeiro
* and
Alexsandro A. Ferreira
Departamento de Estatística, Universidade Federal de Pernambuco, Cidade Universitária, Recife 50670, Brazil
*
Author to whom correspondence should be addressed.
Stats 2025, 8(1), 21; https://doi.org/10.3390/stats8010021
Submission received: 17 January 2025 / Revised: 27 February 2025 / Accepted: 1 March 2025 / Published: 5 March 2025

Abstract

:
In recent years, the gamma-G family has gained significant popularity for generating new distributions by incorporating an additional shape parameter into the baseline model. This article compiles and documents the existing gamma-G distributions and demonstrates their application to real COVID-19 data.

1. Introduction

Classical distributions are widely used across various fields, but their extended forms are often necessary for more flexible modeling of real-world data. The literature has witnessed a surge in generalized distributions by incorporating additional parameters in baseline ones. Notable generators include Marshall–Olkin-G  [1], Beta-G [2], and gamma odd Burr X-G  [3]. This has resulted in numerous new distributions, some of which, like the exponentiated Weibull [4], beta flexible Weibull [5], and Kumaraswamy generalized Lomax [6], are now well-established.
The gamma-G family [7,8] has been extensively studied by investigating its mathematical properties for any baseline distribution [9,10]. This work compiles existing distributions in this family, citing their names and authors (see references for details), aiming to inspire further research into unexplored gamma-G distributions.
The gamma-G family of distributions occupies a significant place in the statistical literature due to its flexibility in modeling diverse datasets. The emergence of more than forty distributions highlights its importance. In addition, their densities can be expressed as linear combinations of exponentiated-G densities, making their main properties easy to obtain [9,10,11,12]. Thus, this combination of flexibility and analytical ease makes the gamma-G family an important tool for theoretical and applied statistical research.
The article proceeds as follows. Section 2 gives an overview of the gamma-G family. Section 3 examines its dual family and addresses related works. Section 4 presents simulations based on the two families. Section 5 applies both families to the COVID-19 data. Finally, Section 6 concludes the paper.

2. The Gamma-G Family

Ref. [7] defined the gamma-G (“ZB-G”) generator, whose probability density function (PDF) and cumulative distribution function (CDF) are
f ( x ) = g ( x ) Γ ( a ) { log [ 1 G ( x ) ] } a 1 ,
and
F ( x ) = γ ( a , log [ 1 G ( x ) ] ) Γ ( a ) ,
respectively, where G ( x ) is any baseline CDF, g ( x ) = d G ( x ) / d x , x I R , γ ( a , z ) = 0 z t a 1 e t d t is the lower incomplete gamma function, Γ ( · ) is the gamma function, and a > 0 is a shape parameter. For a = 1 , Equation (1) reduces to the baseline density. The hazard rate function (HRF) is derived from Equations (1) and (2) as h ( x ) = f ( x ) / [ 1 F ( x ) ] .
The ZB-modified Weibull (ZB-MW) [13], an example of the ZB-G family, uses the CDF G ( x ) = 1 exp { α x β exp ( λ x ) } , where α > 0 is a scale parameter and λ and β are shape parameters. Consequently, its PDF and HRF (for x > 0 ) are
f ( x ) = 1 Γ ( a ) α x β 1 ( β + λ x ) { α x β exp ( λ x ) } a 1 exp { λ x α x β exp ( λ x ) } ,
and
h ( x ) = α x β 1 ( β + λ x ) { α x β exp ( λ x ) } a 1 exp { λ x α x β exp ( λ x ) } Γ ( a ) γ [ a , α x β exp ( λ x ) ] .
The four-parameter ZB-MW model includes, as special cases, the gamma Weibull, gamma exponential, gamma modified exponential, gamma modified Rayleigh, exponential, Rayleigh, and Weibull distributions. Figure 1a displays potential density shapes, suitable for right-skewed data. Figure 1b reports the HRF with four shapes: increasing, decreasing, bathtub, and unimodal.

3. The Dual Gamma Generalized Family

Similarly, ref. [8] introduced the Ristić–Balakrishnan-G (“RB-G”) family, with PDF defined as (for x I R )
f ( x ) = g ( x ) Γ ( a ) { log [ G ( x ) ] } a 1 .
The RB-Burr XII (RB-BXII) distribution [14] comes from the BXII CDF G ( x ) = 1 [ 1 + ( x / s ) c ] d , where s > 0 is a scale and d , c > 0 are shape parameters, whose PDF and HRF (for x > 0 ) are
f ( x ) = c d x c 1 s c Γ ( a ) 1 + x s c ( d + 1 ) log 1 1 + x s c d a 1 ,
and
h ( x ) = c d x c 1 1 + x s c ( d + 1 ) s c γ a , log 1 1 + x s c d log 1 1 + x s c d a 1 .
Figure 2a,b depict the PDF and HRF of the RB-BXII model for some parameters, which show various forms. The ZB-G and RB-G families include the baseline parameters plus an extra shape parameter. Table 1 lists thirty-six members of these classes.
The properties of both families are derived from the exponentiated-G class, as detailed by [11] for ZB-G and [12] for RB-G. Thus, the main properties, such as moments and generating function, are derived from the exponentiated baseline distribution. The random number generation method is detailed in [9] in the form (for 0 < u < 1 )
F 1 ( u ) = G 1 1 exp Q 1 a , 1 u
where Q 1 ( a , u ) is the inverse function of Q ( a , x ) = 1 γ ( a , x ) / Γ ( a ) and G 1 ( · ) is the inverse CDF of any baseline distribution. The results in Section 4 are obtained in this way.

4. Simulation

A simulation study is conducted to assess the consistency of the estimates in the models exemplified for each family (ZB-MW and RB-BXII). Random samples of sizes n = 50 , 100, 200, and 500 are generated in three different scenarios. One thousand Monte Carlo replicates are performed, and the average estimates (AEs), biases, and mean squared errors (MSEs) are calculated. The log-likelihood functions for θ = ( a , α , β , λ ) and ζ = ( a , c , d , s ) are optimized using the Nelder–Mead numerical method within the Optim function of the R statistical software 4.4.1, with the actual values of the parameters as initial inputs.
In order to detail the process of the results obtained in Table 2 and Table 3, the case of the ZB-MW distribution follows, and the RB-BXII case is similar. So, the quantile function (qf) of the ZB-MW follows from (3) as (for 0 < u < 1 )
Q ( u ) = β λ W λ α 1 Q 1 ( a , 1 u ) 1 / β β ,
where the function W ( · ) is the log product function. The R script below generates random variates from the ZB-MW model.
require(gsl)  # Load the gsl package
RZBMW <- function(n, \alpha, \beta, \lambda, a) {
u <- runif(n, 0, 1)  # Generate n random uniform values between 0 and 1
A1 <- qgamma(1 - u, a, 1)  # Compute the quantile of the gamma distribution
A2 <- (lambda / beta) ∗ ((A1 / alpha)^(1 / beta)) qf from MW distribution
Q1 <- (beta / lambda) ∗ lambert_W0(A2)  qf from ZB-MW distribution
return(Q1)  # Return the result Q1
Next, an R function is developed that performs Monte Carlo simulations, optimizing the log-likelihood function (by taking the log of the corresponding PDF) using the Nelder–Mead method and calculating the AEs, biases, and MSEs for the parameters. The full script for this section can be found at this link: https://github.com/alexaaf31/gamma_G-family-/blob/main/Gamma-G_simulations.R (accessed on 1 January 2025).
Table 2 shows that the ZB-MW estimators are consistent. As the sample size grows, the AEs converge to true values, and biases and MSEs decrease. Variations in biases and MSEs suggest that parameter choice influences accuracy.
As expected for the RB-G family, the RB-BXII distribution also provides consistent estimators. As the sample size increases, the AEs get closer to the true parameter values, and the biases and MSEs tend towards zero, as shown in Table 3.

5. Application

The dataset reflects the lifetime (in days) of 103 patients diagnosed with COVID-19. It was compiled by a collaborative team of researchers and doctors from Qatar University in Doha, Qatar, and Dhaka Medical College Hospital in Bangladesh. Data collection took place between 12 April and 31 August 2020, at Dhaka Medical College Hospital, with the approval of the hospital’s Ethics Committee. Lifetimes were measured from the date of admission to the hospital until the time of discharge or death of the patient.
The dataset, available at https://www.kaggle.com/tawsifurrahman/covid19-complete-blood-count-clinical-database (accessed on 1 September 2021), is as follows: 61, 77, 17, 72, 71, 9, 190, 69, 98, 7, 36, 128, 128, 157, 128, 36, 127, 5, 126, 34, 122, 121, 213, 152, 152, 39, 39, 1, 243, 91, 91, 60, 30, 60, 23, 1, 151, 31, 115, 61, 183, 122, 26, 30, 46, 108, 136, 136, 8, 23, 23, 112, 6, 6, 13, 83, 13, 7, 53, 23, 12, 6, 36, 65, 95, 42, 16, 13, 14, 131, 41, 48, 137, 75, 9, 14, 14, 8, 14, 14, 7, 7, 7, 21, 14, 9, 14, 14, 18, 67, 14, 14, 14, 14, 14, 19, 13, 20, 13, 45, 14, 14, 14.
Table 4 provides statistical measures for the data, which have positive skewness and kurtosis. The mean is higher than the median and close to the standard deviation (SD), while the amplitude and variance are comparatively high.
The BXII, MW, and log-logistic (LL) distributions serve as baselines in the ZB-G and RB-G families for comparative analysis. The best models are selected from the Cramér–von Mises ( W * ), Anderson–Darling ( A * ) , Akaike information criterion (AIC), consistent AIC (CAIC), Bayesian IC (BIC), and Hannan–Quinn IC (HQIC). Lower values indicate a better fit. Maximum likelihood estimates (MLEs) and goodness-of-fit statistics are calculated using the AdequacyModel script in R.
Table 5 shows the MLEs and their standard errors (SEs) in parentheses. Notably, the ZB-MW, RB-MW, MW, RB-LL, and BXII distributions exhibit higher SE values. In contrast, the ZB-BXII, RB-BXII, RB-LL, and LL distributions provide accurate estimates.
Table 6 shows the RB-LL distribution with the lowest AIC, CAIC, BIC, and HQIC values, whereas the ZB-BXII distribution has the lowest W * and A * statistics. As these models are not nested, the generalized likelihood ratio (LR) test [44] can be applied from
T L R , N N = 1 n i = 1 n log f ( y i | x i , θ ^ ) g ( y i | x i , γ ^ ) 1 n i = 1 n log f ( y i | x i , θ ^ ) g ( y i | x i , γ ^ ) 2 1 n i = 1 n log f ( y i | x i , θ ^ ) g ( y i | x i , γ ^ ) 2 .
Equation (4) converges to a standard normal distribution under the null hypothesis of model equivalence, which is not rejected if | T L R , N N | z α / 2 . At significance level α , the null hypothesis is rejected in favor of F θ over G γ if T L R , N N > z α (or T L R , N N < z α ). For the ZB-GBXII ( f ( y i | x i , θ ) ) and RB-GLL ( g ( y i | x i , γ ) ) models, the test statistic is T L R , N N = 6.3725 . Since T L R , N N < 1.96 , the null hypothesis is rejected at the level of 0.05 in favor of the RB-GLL distribution.
Figure 3a displays estimated densities, and Figure 3b shows estimated and empirical cumulative functions for the top models. The ZB-BXII, RB-BXII, RB-LL, and LL distributions fit well, but the RB-LL distribution fits the data best. Thus, f ^ ( x ) from the RB-LL distribution, obtained by substituting the MLEs of the parameters into its PDF, can accurately estimate various quantities of interest. For example, consider X as a random variable representing the lifetime of individuals with COVID-19. The probability of X falling within an arbitrary interval [ a , b ] can be estimated by a b f ^ ( x ) d x , while the probability of X being less than a certain value c is given by 0 c f ^ ( x ) d x . Furthermore, the mean can be estimated by 0 x f ^ ( x ) d x , and so on.
All previous results are obtained using the AdequacyModel package in R with the Broyden–Fletcher–Goldfarb–Shanno (BFGS) numerical method. The R script is available at https://github.com/alexaaf31/gamma_G-family-/blob/main/Gamma-G%20Application.R (accessed on 1 January 2025).

6. Conclusions

This article briefly surveyed the gamma-G families pioneered by [7,8]. It provided some of their main functions, a few special cases, and an application for COVID-19 data. A list of published articles on the gamma-G family was also presented. It was noted that many distributions remain to be generated compared to other families, such as Beta-G [2] and Marshall–Olkin-G [1].

Author Contributions

Conceptualization, A.A.F. and G.M.C.; methodology, A.A.F. and G.M.C.; software, A.A.F. and G.M.C.; validation, A.A.F. and G.M.C.; formal analysis, A.A.F. and G.M.C.; investigation, A.A.F. and G.M.C.; resources, A.A.F. and G.M.C.; data curation, A.A.F. and G.M.C.; writing—original draft preparation, G.M.C.; writing—review and editing, A.A.F. and G.M.C. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Stated in the text.

Acknowledgments

We thank the editor and reviewers for their valuable feedback. This work was supported by the Fundação de Amparo à Ciência e Tecnologia do Estado de Pernambuco (FACEPE) [IBPG-1448-1.02/20].

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Plots of the PDF (a) and HRF (b) of the ZB-MW distribution.
Figure 1. Plots of the PDF (a) and HRF (b) of the ZB-MW distribution.
Stats 08 00021 g001
Figure 2. Plots of the PDF (a) and HRF (b) of the RB-BXII distribution.
Figure 2. Plots of the PDF (a) and HRF (b) of the RB-BXII distribution.
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Figure 3. (a) Estimated densities of the ZB-BXII, RB-BXII, and RB-LL; (b) estimated and empirical cumulative functions of these models.
Figure 3. (a) Estimated densities of the ZB-BXII, RB-BXII, and RB-LL; (b) estimated and empirical cumulative functions of these models.
Stats 08 00021 g003
Table 1. Distributions in the gamma-G family.
Table 1. Distributions in the gamma-G family.
No.DistributionAuthor(s)
1.Gamma exponentiated exponential[8]
2.Gamma exponentiated Weibull[15]
3.Gamma Pareto[16]
4.Gamma uniform[17]
5.Gamma extended Fréchet[18]
6.Gamma half normal[19]
7.ZB-LL[20]
8.Gamma linear failure rate[21]
9.Gamma Dagum[22,23]
10.Gamma normal[24]
11.Gamma generalized Weibull[11]
12.Gamma log-normal[9]
13.Gamma Weibull[9,10]
14.Gamma Gumbel[9,10]
15.ZB-MW[13]
16.Gamma half-Cauchy[25]
17.Gamma logistic[10]
18.Gamma exponentiated exponential Weibull[26]
19.Gamma Burr type X[27]
20.Gamma Maxwell[28]
21.Gamma log-logistic Weibull[29]
22.RB-BXII[14]
23.Gamma Burr III[30]
24.Gamma Slash[31]
25.Gamma extended exponential[32]
26.Gamma Lindley–Poisson[33]
27.Gamma Kumaraswamy[34]
28.Gamma Rayleigh[35]
29.Gamma power half-logistic[36]
30.Gamma log-logistic Erlang truncated exponential[37]
31.Gamma generalized Maxwell[38]
32.Gamma generalized Lindley log-logistic[39]
33.Gamma power Lindley[40]
34.Gamma Chen[41]
35.Gamma Lindley[42]
36.Gamma Gompertz[43]
Table 2. Simulations from the ZB-MW distribution.
Table 2. Simulations from the ZB-MW distribution.
(0.9, 0.6, 1.2, 1.5)(1.9, 0.4, 0.8, 2.5)(2.0, 0.2, 0.2, 0.6)
n θ AE Bias MSE AE Bias MSE AE Bias MSE
50a0.97280.07281.03072.09570.41973.84202.10330.10330.6619
α 1.04500.44502.06601.23530.83534.24780.24800.04800.0501
β 2.31061.110612.81332.34741.547414.75440.48170.28170.3140
λ 1.4252−0.07471.55782.2546−0.24532.10230.5540−0.04590.0449
100a0.93040.03040.51101.95300.05302.17562.04750.04750.5442
α 0.83910.23910.65810.80940.40941.15100.23870.03870.0427
β 1.94810.74814.52131.71200.91204.44320.32220.12220.0972
λ 1.4085−0.09140.81802.3235−0.17641.24090.5987−0.00120.0230
200a0.8961−0.00380.29951.8995−0.00041.15421.9906−0.00930.4053
α 0.66180.06180.13590.53450.13450.19000.22010.02010.0318
β 1.64630.44631.32291.30820.50821.75150.25110.05110.0205
λ 1.4269−0.07300.25202.3528−0.14710.44070.61610.01610.0106
500a0.8995−0.00040.26671.90020.00021.00552.01730.01730.3730
α 0.65090.05090.09470.49730.09730.11340.22430.02430.0275
β 1.58540.38541.09461.22900.42901.30540.23840.03840.0164
λ 1.4175−0.08240.17642.3720−0.12790.29320.61130.01130.0083
Table 3. Simulations from the RB-BXII distribution.
Table 3. Simulations from the RB-BXII distribution.
(0.9, 2.5, 0.1, 0.2)(0.5, 7.0, 0.3, 0.4)(1.5, 3.2, 0.8, 0.1)
n ζ AE Bias MSE AE Bias MSE AE Bias MSE
50a1.17450.27450.82290.58670.08670.30933.41711.917120.8649
c3.53011.030110.84469.07032.070336.07334.00260.80262.2241
d0.12840.02840.01470.34960.04960.04812.10721.307224.2399
s0.31190.11190.09910.43080.03080.13420.21130.11130.0939
100a1.07750.17750.49300.58200.08200.15472.55761.05768.4377
c3.12300.62304.53248.18061.180617.00913.61600.41600.9437
d0.11620.01620.00850.31470.01470.02401.14180.34183.5938
s0.25460.05460.02990.41220.01220.00250.14610.04610.0250
200a0.95560.05560.20810.56500.06500.07881.88700.38700.9241
c2.73970.23971.23287.39360.39362.45433.37880.17880.2119
d0.11270.01270.00400.30080.00080.01420.7529−0.04700.1796
s0.21430.01430.00460.40550.00550.00070.10690.00690.0008
500a0.92650.02650.14130.55180.05180.05961.82670.32670.5674
c2.61200.11200.27137.22730.22731.25893.34510.14510.1435
d0.11330.01330.00340.30210.00210.01060.7453-0.05460.1428
s0.20840.00840.00250.40450.00450.00050.10470.00470.0001
Table 4. Descriptive statistics.
Table 4. Descriptive statistics.
MeanMedianSDVarianceSkewnessKurtosisMin.Max.
55.66031.00054.8383007.31.1803.6121243
Table 5. MLEs and their SEs for some fitted models to COVID-19 data.
Table 5. MLEs and their SEs for some fitted models to COVID-19 data.
ModelMLEs (SEs)
ZB-BXII ( a , c , k , s ) 16.232 0.279 30.869 131.794
( 0.819 )( 0.022 )( 1.028 )( 1.195 )
RB-BXII ( a , c , k , s ) 0.106 0.985 63.364 315.082
( 0.010 )( 0.002 )( 0.014 )( 0.014 )
ZB-MW ( a , α , β , λ ) 1.043 0.019 0.001 0.963
( 0.796 )( 0.030 )( 0.002 )( 0.176 )
RB-MW ( a , α , β , λ ) 1.286 0.052 0.001 0.794
( 0.527 )( 0.069 )( 0.001 )( 0.231 )
ZB-LL ( a , c , s ) 1.082 28.404 1.432
( 0.499 )( 20.651 )( 0.172 )
RB-LL ( a , c , s ) 9.169 1115.857 2.573
( 1.228 )( 16.287 )( 0.354 )
BXII ( c , k , s ) 1.074 6.037 262.562
( 0.107 )( 5.965 )( 300.345 )
MW ( α , β , λ ) 0.021 0.001 0.955
( 0.008 )( 0.002 )( 0.117 )
LL ( c , s ) 1.456 32.076
( 0.116 )( 3.868 )
Table 6. Goodness-of-fit statistics for the fitted models to COVID-19 data.
Table 6. Goodness-of-fit statistics for the fitted models to COVID-19 data.
Model W * A * AICCAICBICHQIC
ZB-BXII0.2601.663 1041.918 1042.326 1052.457 1046.187
RB-BXII 0.326 1.945 1039.501 1039.909 1050.040 1043.770
ZB-MW 0.373 2.168 1040.769 1041.177 1051.308 1045.038
RB-MW 0.387 2.247 1041.798 1042.206 1052.337 1046.067
ZB-LL 0.327 2.029 1045.631 1045.874 1053.536 1048.833
RB-LL 0.313 1.858 1036.3931036.6351044.2971039.594
BXII 0.368 2.177 1040.784 1041.026 1048.688 1043.985
MW 0.386 2.245 1039.813 1040.055 1047.717 1043.014
LL 0.334 2.065 1043.644 1043.764 1048.913 1045.778
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Cordeiro, G.M.; Ferreira, A.A. The Gamma-G Family: Brief Survey and COVID-19 Application. Stats 2025, 8, 21. https://doi.org/10.3390/stats8010021

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Cordeiro GM, Ferreira AA. The Gamma-G Family: Brief Survey and COVID-19 Application. Stats. 2025; 8(1):21. https://doi.org/10.3390/stats8010021

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Cordeiro, Gauss M., and Alexsandro A. Ferreira. 2025. "The Gamma-G Family: Brief Survey and COVID-19 Application" Stats 8, no. 1: 21. https://doi.org/10.3390/stats8010021

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Cordeiro, G. M., & Ferreira, A. A. (2025). The Gamma-G Family: Brief Survey and COVID-19 Application. Stats, 8(1), 21. https://doi.org/10.3390/stats8010021

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