# Gompertz Ampadu Class of Distributions: Properties and Applications

Authors: Abukari Alhassan
Journal: Journal of Probability and Statistics
 (2022)

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## Abstract

This paper introduces a new generator family of distributions called the Gompertz Ampadu-G family. Based on the generator, the Lomax distribution was modified into Gompertz Ampadu Lomax. The new distribution has a flexible hazard rate function that has upside-down and bathtub shapes, including increasing and decreasing hazard rate functions. The distribution comes with some desirable statistical properties. The distribution is applied to real-life data. Parameter estimates and test statistics show a better fit for the competitive models.


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## Body

## 1. Introduction

Several classical distributions have been modified over the last decade to cater to improved modeling in the fields of insurance, economics, actuarial, engineering, environmental, medical, biological studies, demography, sciences, and finance. In the attempt to develop these classical distributions, several generator families of distribution become relevant and are used to modify existing distributions to make them more flexible. Some generator families of distributions that have been employed over the years are: Beta-G by Eugene et al., [1]; Marshall–Olkin generated family (MO-G) by Marshall and Olkin [2]; proportional reversed hazard rate family by Gupta and Gupta [3]; gamma-G (type 1) by Zografos and Balakrishanan [4]; logistic-G by Torabi and Montazari [5]; Transformed-Transformer (T-X) by [6] and Lomax-G by Cordeiro et al., [6]. Some other G family of distributions include the new generalized two-sided class of distributions with emphasis on two-sided generalized normal distribution by Kharazm and Zargari [7]; a new family of the continuous distributions, the extended Weibull-G family by Mustafa et al. [8]; the odd power Lindley generator of probability distributions by Mustafa et al. [9]; the HJORTH’S IDB generator of distributions by Mustafa et al. [10]; a new flexible family of continuous distributions by Altun et al. [11], the type I quasi Lambert family by Hamedani et al. [12] and the Fréchet Topp Leone-G Family of Distributions by Hesham et al. [13].Most of these distributions have a few parameters, which restrict modeling flexibilities. In this study, the Gompertz generator family [14] and that of the Ampadu-generator family [15] are mutually improved to make them more flexible. The two distributions are combined to create three parameters. Unlike Alizadeh et al. [16] with only one scale parameter and Tahir et al. [17] with only two-scale parameters, the Korkmaz et al. [18] model also had the advantage of being capable of modeling various shapes but was only applicable in aging and failure criteria. The Gomperz–Ampadu has three scale parameters that have more power in modeling data with highly dispersed characteristics.The GompertzG family CDF is given as(1)Fx=1−eα/β1−1−Hx:τ−βα>0,β>0,x>0,where Hx and hx are CDF and the PDF of the baseline distribution. The Gompertz G family has two parameters α and β . Differentiating (1) gives the PDF of Gompertz G distribution as(2)fx:α,β,τ=αhx:τ1−Hx:τ−β−1eα/β1−1−Hx:τ−β.The AmpaduG family CDF is given as(3)Hx:γ,τ=1−e−γGx:τ21−e−γ,γ>0,x>0,where Gx and hx are the CDF and the PDF of the baseline distribution. The Ampadu G has one scale parameter without any shape parameter. This means that the distribution is limited with the control of skewness of which shape parameters give.Differentiating (3) gives the PDF of the Ampadu G family as(4)hx:γ,τ=2γgx:τGx:τe−γGx:τ21−e−γ.


## 2. The Gompertz–Ampadu-Generator Family

The absence of the shape parameter in AmpaduG motivates combining it with Gompertz G, which has a shape parameter. This arrives at the Gompertz Ampadu G, which has both shape and scale parameters. Combining (1) and (3), we obtain the Gompertz Ampadu-generator family of distributions. In this regard, equation (3) is inserted in (1) to get the CDF of the Gompertz Ampadu-G family as(5)Fx=1−eα/β1−1−1−1−e−γGx:τ2/1−e−γ−β,α>0,β>0,γ>0,x>0.The Gompertz AmpaduG has three parameters, thus, two-scale parameters α and γ with one shape parameter β.Hence, (5) is differentiated to get the PDF of the Gompertz Ampadu-G family as(6)fx:α,β,τ=2αγgx:τGx:τe−γGx:τ21−e−γ−11−e−γ−1−e−γGx:τ21−e−γ−β−1.The hazard function of the Gompertz Ampadu-generator is(7)dx=2αγgx:τGx:τe−γGx:τ21−e−γ−11−e−γ−1−e−γGx:τ2/1−e−γ−β−11−1−eα/β1−1−1−1−e−γGx:τ2/1−e−γ−β.


## 3. Some Statistical Properties

Finding the inverse of (5), we obtain the quantile of Gompertz Ampadu-generator as(8)xu=G−1−1γln1+1−e−γ1−1−βαlnu+1−1/β1/2,α>0,β>0,γ>0,x>0.The quantile function generates random numbers to be used for the simulation exercise. This is about inverting the CDF of the Gompertz AmpaduG.


### 3.1. Useful Expansion

It can be shown that the CDF of the Gompertz Ampadu-G family depicts the partial expansion. The partial expansion helps in the derivation of the other statistical properties like moments, entropy, stress reliability, and more [18]. Given that Gx:τ is the baseline distribution of any model. The expansion is as follows:(9)Fx=1−∑i=0∞∑j=0i∑k=0∞cijkGx:τ=1−∑k=0∞vkGx:τ=∑k=0∞zkGx:τ,cijk=−11+ki!ij−jykαβi,vk=∑i=0∞∑j=0∞cijk,z0=1−v0 and zk=−vk for k=1,2,… With this Fkx=Gxk, which stands for exponentiated G cumulative with power parameter k>0. Some of the structural properties of the exponentiated G are shown by Mudholkar et al. [19]; Gupta and Kundu [20] and many other authors. The expansion will serve a good purpose for other desirable properties of the Gompertz Ampadu-generator to be further developed.


## 3.1. Useful Expansion

It can be shown that the CDF of the Gompertz Ampadu-G family depicts the partial expansion. The partial expansion helps in the derivation of the other statistical properties like moments, entropy, stress reliability, and more [18]. Given that Gx:τ is the baseline distribution of any model. The expansion is as follows:(9)Fx=1−∑i=0∞∑j=0i∑k=0∞cijkGx:τ=1−∑k=0∞vkGx:τ=∑k=0∞zkGx:τ,cijk=−11+ki!ij−jykαβi,vk=∑i=0∞∑j=0∞cijk,z0=1−v0 and zk=−vk for k=1,2,… With this Fkx=Gxk, which stands for exponentiated G cumulative with power parameter k>0. Some of the structural properties of the exponentiated G are shown by Mudholkar et al. [19]; Gupta and Kundu [20] and many other authors. The expansion will serve a good purpose for other desirable properties of the Gompertz Ampadu-generator to be further developed.


## 4. Estimation

The maximum likelihood estimates (MLEs) of the parameters of a new family of distributions provide desirable properties [21]. In this new family of distributions, assuming x1,x2,…,xn are the observed values taken from the Gompertz Ampadu generator with parameters ϖ,α,β,γ and τ, let ϖ=α,λ,τT represent the r×1 parameter vector. Hence the total log-likelihood function for ϖ is given by(10)ℓn=ℓnϖ=nlog+∑i=1nloggxi;τ−β+1∑i=1nlog1−1−1−e−γGx:τ21−e−γ+αβ∑i=1nlog1−1−1−1−e−γGx:τ21−e−γ−β.In respective ofα,β,γ, and τ, the mathematical software was used to differentiate the total log function to get the score function and estimate for the parameter values.


## 5. Application of Gompertz–Ampadu Family to Lomax Distribution

The Gompertz AMPADU-generator was applied on the Lomax distribution of CDFGx=1−1+λx−θ,x>0,λ>0,θ>0, and PDF: gx=θλ1+θx−θ−1,x>0 to obtain Gompertz Ampadu Lomax distribution (GA_L). The CDF, PDF, and hazard functions of the new distribution respectively follow as(11)Fx=1−eα/β1−1−1−1−e−γ1−1+λx−θ2/1−e−γ−β,α>0,β>0,γ>0,λ>0,θ>0,x>0.Hence, (5) is differentiated to get the PDF of the Gompertz Ampadu-G family as(12)fx:α,β,τ=2αγθλe−γ1−1+λx−θ21−1+λx−θ1−e−γ1+θxθ+11−e−γ−1−e−γ1−1+λx−θ21−e−γ−β−1,α>0,β>0,γ>0,λ>0,θ>0,x>0Some selected parameter values of the GA_L, whose shapes are presented in Figure1 as follows, show more flexible shapes. The flexible nature of the distribution means that it can model data that is right-skewed, left-skewed, and symmetrical.Figure 1
PDF plot of GA_L distribution.The hazard(H) function of the GA_L distribution is(13)dx:α,β,τ=2αγθλe−γ1−1+λx−θ21−1+λx−θ/1−e−γ1+θxθ+11−e−γ−1−e−γ1−1+λx−θ2/1−e−γ−β−11−1−eα/β1−1−1−1−e−γ1−1+λx−θ2/1−e−γ−β,α>0,β>0,γ>0,λ>0,θ>0,x>0.The hazard function can be used to model the failure rate of a system. The hazard plot of some parameter values of the hazard function is displayed in Figure2 as follows. The GA_L shows increasing and decreasing bathtub shapes.Figure 2
Hazard plot of some parameter values for GA_L.The quantile function of the GA_L distribution is(14)xu=1−1+λx−θ−1−1γln1+1−e−γ1−1−βαlnu+1−1/β1/2,α>0,β>0,γ>0,λ>0,θ>0,x>0.The quantile function was used to generate random numbers for the simulation exercise. The function was derived by inverting the CDF of the GA_L distribution.


## 6. Simulation Study

In assessing the performance of the MLEs, this was done through the Monte Carlo simulation. In this case, 1000 replications were performed using theR software (stats4 package). The results in Table 1 show decreasing standard errors (SE) and root mean square errors (RMSE) with increasing sample sizes.Table 1
Monte Carlo simulation results of GA_L.
True parameter valueABIASSRMSEαβγλθ^nα^β^γ^λ^θ^α^β^γ^λ^θ^0.30.30.40.50.2504.2670.2350.0270.3560.0274.7860.1431.4400.3792.4431003.9730.2300.0260.3240.0265.5520.2360.4780.3391.3781503.3240.2530.0210.3020.0215.1220.2280.8220.3402.7222003.2400.3210.0210.2950.0213.0840.2260.2900.3051.1915003.2400.3210.0210.2950.0213.0840.2260.2900.3051.19110000.2400.0010.0010.0950.0011.0840.0060.0900.0050.0910.30.30.20.50.2505.3020.3250.0290.0320.0213.0580.3540.1960.2601.0961004.5070.3090.0270.0410.0085.8100.3320.1820.2421.0821504.2000.3010.0460.0230.0365.4720.3210.2540.2370.2552003.8920.3880.0540.0250.0355.1150.3080.2330.2680.2335003.0920.0880.0440.0150.0255.0150.2080.1330.1680.13310000.0920.0080.0040.0050.0054.0150.0080.0330.0680.0330.60.30.30.40.1501.0610.1130.0490.0120.0385.3370.3590.2560.2030.3561001.0210.1110.0350.0120.0265.1670.3450.2890.2030.3891501.0120.1080.0300.0120.0215.0870.3360.1310.2020.3332001.0060.1100.0310.0120.0225.0660.4390.3740.3020.3745001.0050.0100.0210.0100.0125.0560.3390.2740.2020.27410000.0050.0000.0010.0000.0022.0560.0390.0740.0020.0740.30.10.40.20.5504.0020.1520.0880.0470.0793.8530.4670.1430.4540.6431002.2770.1260.1910.0250.1823.2050.5480.3120.3333.3131501.6970.1150.0450.0190.0363.5480.5360.1160.5234.1162001.3360.1880.0370.0420.0283.0280.5260.2270.5165.1275001.0360.0880.0270.0320.0183.0180.4260.1270.4165.02710000.0360.0080.0070.0020.0081.0180.0060.0270.0162.027Values in bold are the true parameter values of the GA_L.


## 7. Application of Gompertz–Ampadu–Lomax to Myelogenous Leukaemia Data

This study used the survival times data in the weeks of 33 patients suffering from acute myelogenous leukaemia. The data was first used by Feigl and Zelen [22]. The dataset is represented in Table 2; Table 3 gives the estimates and standard deviations of the different models used; and Table 4 also gives the AIC, BIC, Kolmogorov–Smirnov test, and the p-values. The results show smaller AIC, BIC, and Kolmogorov–Smirnov values than the competing models. This means that the Gompertz Ampadu Lomax is better than the Gompertz Lomax, Ampadu Lomax, and the Lomax itself for modeling survival data.Table 2
Survival times of acute myelogenous leukaemia data.
65156100134161081214391435626221156556651771622342384330443Table 3
The estimates and standard errors (in parentheses) for survival times.
EstimatesModelαβγλθGA_L21.23601.24381.03880.10420.5467(0.3494)(4.9349)(0.1233)(0.0122)(0.2121)Ampadu_L1.09610.03121.2483(6.7061)(0.3646)(1.6000)Gompertz_L11.40143.98960.01510.0506(8.51× 10−1)(1.08× 10−1)(6.10× 10−2)(2.56× 10−3)L1.30900.0175(2.0548)(0.0054)Table 4
Goodness-of-fit for dataset survival times.
Model-LAICBICK–SPvalueGA_L139.2600308.0033315.59590.20730.7415Ampadu_L143.4200314.0132320.01940.23650.4701Gompertz_L142.4200311.0141315.00440.22880.5446L143.6700323.1425316.63240.23660.4697Figure3 gives the goodness-of-fit histogram plots of the competing models with the survival data. The plot shows Gompertz Ampadu Lomax (GA_L) to be better than the other models.Figure 3
Histogram plot of models with survival times data.


## 8. Conclusion

The Gompertz Ampadu-G family is derived in this study. Based on the generator, the Lomax distributions were also modified into Gompertz Ampadu Lomax. The new distribution has a flexible hazard rate function, which has upside-down and bathtub shapes, including increasing, decreasing hazard rate functions. The distribution comes with some desirable statistical properties and is applied to a real-life data. Parameter estimates and test statistics show a better fit than competitive models. However, it is recommended that the Gompertz Ampadu-generator family of distributions be used to develop classical distributions like Weibull, exponential, Fréchet, and many more distributions.



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