# A New Generalization of the Lomax Distribution with Increasing, Decreasing, and Constant Failure Rate

Authors: Pelumi E. Oguntunde; Mundher A. Khaleel; Mohammed T. Ahmed; Adebowale O. Adejumo; Oluwole A. Odetunmibi
Journal: Modelling and Simulation in Engineering
 (2017)

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## Abstract

Developing new compound distributions which are more flexible than the existing distributions have become the new trend in distribution theory. In this present study, the Lomax distribution was extended using the Gompertz family of distribution, its resulting densities and statistical properties were carefully derived, and the method of maximum likelihood estimation was proposed in estimating the model parameters. A simulation study to assess the performance of the parameters of Gompertz Lomax distribution was provided and an application to real life data was provided to assess the potentials of the newly derived distribution. Excerpt from the analysis indicates that the Gompertz Lomax distribution performed better than the Beta Lomax distribution, Weibull Lomax distribution, and Kumaraswamy Lomax distribution.


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## Body

## 1. Introduction

The Lomax distribution can also be called Pareto Type II distribution and its application can be found in many fields like actuarial science, economics, and so on [1]. The distribution was defined by Lomax [2] and it is a heavy-tailed distribution. It has also been considered to be useful in reliability and life testing problems in engineering and in survival analysis as an alternative distribution [3, 4].Modified and extended versions of the Lomax distribution have been studied; examples include the weighted Lomax distribution [4], exponential Lomax distribution [5], exponentiated Lomax distribution [6], gamma Lomax distribution [7], transmuted Lomax distribution [8], Poisson Lomax distribution [9], McDonald Lomax distribution [10], Weibull Lomax distribution [11], and power Lomax distribution [12]. Besides, estimation of the parameters of Lomax distribution under general progressive censoring has also been considered by Al-Zahrani and Al-Sobhi [1].In addition to the generalized families of distributions mentioned earlier, there are several other generalized families of distributions in the literature and these are contained in Owoloko et al. [13], Oguntunde et al. [14], Cordeiro et al. [15], and Alizadeh et al. [16]. Meanwhile, of interest to us in this research is to extend the Lomax distribution using the Gompertz generalized family of distributions due to Alizadeh et al. [16] because it is relatively new; it has not yet been rigorously explored, and it has some potentials which would be revealed in the later part of this article.The rest of this article is therefore organized as follows: in Section2, the densities of the Gompertz Lomax distribution (henceforth, it is referred to as GoLom distribution) are derived; its statistical properties are established including estimation of the unknown parameters; in Section 3, a simulation study was provided to investigate the performances of the unknown parameters; then an application to a real life data was provided, followed by a concluding remark.


## 2. The Gompertz Lomax Distribution

To start with, the cumulative distribution function (cdf) and probability density function (pdf) of the Lomax distribution with parametersα and β are given by(1)Gx=1-1+βx-α;α>0,β>0,(2)gx=αβ1+βx-α+1;α>0,β>0,respectively, where α and β are the shape and scale parameters, respectively.According to Alizadeh et al. [16], the cdf and pdf of the Gompertz generalized family of distribution are given by(3)Fx=1-eθ/γ1-1-Gx-γ;θ>0,γ>0,(4)fx=θgx1-Gx-γ-1eθ/γ1-1-Gx-γ;θ>0,γ>0,where θ and γ are additional shape parameters and their role is to vary tail weights.G
x and gx are the cdf and pdf of the parent (or baseline) distribution, respectively.Now, if the density in (1) in inserted into (3), then the cdf of the GoLom distribution is given by(5)Fx=1-eθ/γ1-1+βxαγ;θ>0,γ>0,α>0,β>0.Its associated pdf is derived by inserting the densities in (1) and (2) into (4) as follows:(6)fx=θαβ1+βxαγ-1eθ/γ1-1+βxαγ;θ>0,γ>0,α>0,β>0,where α, θ, and γ are shape parameters; β is the scale parameter.Plots for the pdf of the GoLom distribution at various selected values are displayed in Figure1.Figure 1
Plot for the pdf of GoLom distribution.Remark 1.
It is clear in Figure1 that the shape of the GoLom distribution could be decreasing or inverted bathtub (depending on the value of the parameters). Also, it could be positively skewed or negatively skewed. Studying the tail behaviour of the GoLom distribution, it can be deduced that the distribution is heavy-tailed.


### 2.1. Expansion for the Densities of GoLom Distribution

Following Alizadeh et al. [16], the density of the GoLom distribution in (5) can be expanded as follows:(7)Fx=1-∑i=0∞∑j=0i∑k=0∞wi,j,kHkx=1-∑k=0∞akHkx,where wi,j,k=-1i+k/i!ij-jγkθ/γi, ak=∑i=0∞∑j=0iwi,j,k, and Hkx is the cdf of the exponentiated Lomax distribution with power k>0.The associated pdf can be expressed as a linear mixture of the exponentiated Lomax function as follows:(8)fx=∑k=0∞bk+1hk+1x,where hk+1=k+1gxGxk and bk=-ak and Gx and gx are the cdf and pdf of the Lomax distribution as defined in (1) and (2), respectively.


### 2.2. Reliability Analysis

The expressions for the reliability function, hazard function (or failure rate), reversed hazard function, and odds function are all derived and established in this subsection.Reliability Function. Reliability or survival function can be obtained from(9)Sx=1-Fx.Therefore, the reliability function of the GoLom distribution is given by(10)Sx=eθ/γ1-1+βxαγ;θ>0,γ>0,α>0,β>0.It is good to note that the shape of the reliability function of GoLom distribution would be a constant when the value of parameter β=0 and α=θ=γ=1. An illustration to this is as shown in Figure 2.Figure 2
Survival function of GoLom distribution atβ=0 and α=θ=γ=1.Hazard Function. Hazard function can be obtained from(11)hx=fxSx.Therefore, the hazard function of the GoLom distribution is given by(12)hx=θαβ1+βxαγ-1;θ>0,γ>0,α>0,β>0.Plots for the hazard function of the GoLom distribution at various selected values are displayed in Figure 3.Figure 3
Plot for the hazard function of GoLom distribution.Remark 2.
It can be deduced from Figure3 that the shape of the hazard function of the GoLom distribution could be constant, increasing, or decreasing (depending on the value of the parameters).Reversed Hazard Function. Reversed hazard function can be derived from(13)rx=fxFx.Therefore, the reversed hazard function for the GoLom distribution is given by(14)rx=θαβ1+βxαγ-1eθ/γ1-1+βxαγ1-eθ/γ1-1+βxαγ;θ>0,γ>0,α>0,β>0.Odds Function. Odds function can be derived from(15)Ox=FxSx.Therefore, the odds function for the GoLom distribution is given by(16)Ox=1-eθ/γ1-1+βxαγeθ/γ1-1+βxαγ;θ>0,γ>0,α>0,β>0.


### 2.3. Quantile Function and Median

Quantile function can be derived from(17)Qu=F-1u.Therefore, the quantile function of the GoLom distribution is given by(18)Qu=β-11-γθlog⁡1-u1/αγ-1,where u~Uniform0,1.Random numbers can be generated from the GoLom distribution using(19)x=β-11-γθlog⁡1-u1/αγ-1.The median of the GoLom distribution can be derived by substituting u=0.5 into (18) as follows:(20)Median=β-11-γθlog⁡0.51/αγ-1.Other quartiles can also be derived from (18) by substituting the appropriate value of “u.”


### 2.4. Distribution of Order Statistics

Letx1,x2,…,xn be a random sample from a cdf and pdf of a Gompertz Lomax distribution as defined in (5) and (6), respectively; the pdf of the jth order statistics of the GoLom distribution is obtained from(21)fj:nx=n!j-1!n-j!fxFxj-11-Fxn-j.Then, the pdf of jth order statistics for the GoLom distribution is(22)fj:nx=n!j-1!n-j!θαβ1+βxαγ-1eθ/γ1-1+βxαγ1-eθ/γ1-1+βxαγj-1eθ/γ1-1+βxαγn-j.Therefore, the distribution of minimum and maximum order statistics for the GoLom distribution is given by(23)f1:nx=nθαβ1+βxαγ-1eθ/γ1-1+βxαγeθ/γ1-1+βxαγn-1,fn:nx=nθαβ1+βxαγ-1eθ/γ1-1+βxαγ1-eθ/γ1-1+βxαγn-1.


### 2.5. Parameter Estimation

The parameters of the GoLom distribution can be estimated using the method of maximum likelihood (MLE) as follows: letx1,x2,…,xn denote random samples each having the pdf of the GoLom distribution; then the likelihood function is given by(24)fx1,x2,…,xn;α,β,γ,θ=∏i=1nθαβ1+βxiαγ-1eθ/γ1-1+βxiαγ.Let l denote the log-likelihood function; that is, let l=log⁡fx1,x2,…,xn;α,β,γ,θ; then(25)l=nlog⁡θ+nlog⁡α+nlog⁡β+αγ-1∑i=1nlog1+βxi+θγ∑i=1n1-1+βxiαγ.Solving dl/dα=0, dl/dβ=0, dl/dγ=0, and dl/dθ=0 simultaneously gives the maximum likelihood estimates of parameters α, β, γ, and θ. Meanwhile, the solution cannot be gotten analytically except numerically when data sets are available. Software like R, MATLAB, MAPLE, and so on could be used to get the estimates.


## 2.1. Expansion for the Densities of GoLom Distribution

Following Alizadeh et al. [16], the density of the GoLom distribution in (5) can be expanded as follows:(7)Fx=1-∑i=0∞∑j=0i∑k=0∞wi,j,kHkx=1-∑k=0∞akHkx,where wi,j,k=-1i+k/i!ij-jγkθ/γi, ak=∑i=0∞∑j=0iwi,j,k, and Hkx is the cdf of the exponentiated Lomax distribution with power k>0.The associated pdf can be expressed as a linear mixture of the exponentiated Lomax function as follows:(8)fx=∑k=0∞bk+1hk+1x,where hk+1=k+1gxGxk and bk=-ak and Gx and gx are the cdf and pdf of the Lomax distribution as defined in (1) and (2), respectively.


## 2.2. Reliability Analysis

The expressions for the reliability function, hazard function (or failure rate), reversed hazard function, and odds function are all derived and established in this subsection.Reliability Function. Reliability or survival function can be obtained from(9)Sx=1-Fx.Therefore, the reliability function of the GoLom distribution is given by(10)Sx=eθ/γ1-1+βxαγ;θ>0,γ>0,α>0,β>0.It is good to note that the shape of the reliability function of GoLom distribution would be a constant when the value of parameter β=0 and α=θ=γ=1. An illustration to this is as shown in Figure 2.Figure 2
Survival function of GoLom distribution atβ=0 and α=θ=γ=1.Hazard Function. Hazard function can be obtained from(11)hx=fxSx.Therefore, the hazard function of the GoLom distribution is given by(12)hx=θαβ1+βxαγ-1;θ>0,γ>0,α>0,β>0.Plots for the hazard function of the GoLom distribution at various selected values are displayed in Figure 3.Figure 3
Plot for the hazard function of GoLom distribution.Remark 2.
It can be deduced from Figure3 that the shape of the hazard function of the GoLom distribution could be constant, increasing, or decreasing (depending on the value of the parameters).Reversed Hazard Function. Reversed hazard function can be derived from(13)rx=fxFx.Therefore, the reversed hazard function for the GoLom distribution is given by(14)rx=θαβ1+βxαγ-1eθ/γ1-1+βxαγ1-eθ/γ1-1+βxαγ;θ>0,γ>0,α>0,β>0.Odds Function. Odds function can be derived from(15)Ox=FxSx.Therefore, the odds function for the GoLom distribution is given by(16)Ox=1-eθ/γ1-1+βxαγeθ/γ1-1+βxαγ;θ>0,γ>0,α>0,β>0.


## 2.3. Quantile Function and Median

Quantile function can be derived from(17)Qu=F-1u.Therefore, the quantile function of the GoLom distribution is given by(18)Qu=β-11-γθlog⁡1-u1/αγ-1,where u~Uniform0,1.Random numbers can be generated from the GoLom distribution using(19)x=β-11-γθlog⁡1-u1/αγ-1.The median of the GoLom distribution can be derived by substituting u=0.5 into (18) as follows:(20)Median=β-11-γθlog⁡0.51/αγ-1.Other quartiles can also be derived from (18) by substituting the appropriate value of “u.”


## 2.4. Distribution of Order Statistics

Letx1,x2,…,xn be a random sample from a cdf and pdf of a Gompertz Lomax distribution as defined in (5) and (6), respectively; the pdf of the jth order statistics of the GoLom distribution is obtained from(21)fj:nx=n!j-1!n-j!fxFxj-11-Fxn-j.Then, the pdf of jth order statistics for the GoLom distribution is(22)fj:nx=n!j-1!n-j!θαβ1+βxαγ-1eθ/γ1-1+βxαγ1-eθ/γ1-1+βxαγj-1eθ/γ1-1+βxαγn-j.Therefore, the distribution of minimum and maximum order statistics for the GoLom distribution is given by(23)f1:nx=nθαβ1+βxαγ-1eθ/γ1-1+βxαγeθ/γ1-1+βxαγn-1,fn:nx=nθαβ1+βxαγ-1eθ/γ1-1+βxαγ1-eθ/γ1-1+βxαγn-1.


## 2.5. Parameter Estimation

The parameters of the GoLom distribution can be estimated using the method of maximum likelihood (MLE) as follows: letx1,x2,…,xn denote random samples each having the pdf of the GoLom distribution; then the likelihood function is given by(24)fx1,x2,…,xn;α,β,γ,θ=∏i=1nθαβ1+βxiαγ-1eθ/γ1-1+βxiαγ.Let l denote the log-likelihood function; that is, let l=log⁡fx1,x2,…,xn;α,β,γ,θ; then(25)l=nlog⁡θ+nlog⁡α+nlog⁡β+αγ-1∑i=1nlog1+βxi+θγ∑i=1n1-1+βxiαγ.Solving dl/dα=0, dl/dβ=0, dl/dγ=0, and dl/dθ=0 simultaneously gives the maximum likelihood estimates of parameters α, β, γ, and θ. Meanwhile, the solution cannot be gotten analytically except numerically when data sets are available. Software like R, MATLAB, MAPLE, and so on could be used to get the estimates.


## 3. Simulation

The behaviour of the parameters of the GoLom distribution was investigated by conducting simulation studies with the aid of R software. Data sets were generated from the GoLom distribution with a replication numberm=1000; random samples of sizes n=25, 50, and 100 were further selected. The simulation was conducted for three (3) different cases using varying true parameter values. The selected true parameter values are α=0.5, β=0.5, γ=0.5, and θ=0.5; α=1, β=1, γ=1, and θ=1; and α=2, β=2, γ=2, and θ=2 for the first, second, and third cases, respectively.The MLE of the true parameters were obtained including the Bias and the Root Mean Square Error (RMSE). The result for the simulation studies is as shown in Tables1, 2, and 3.Table 1
Simulation study atα=0.5, β=0.5, γ=0.5, and θ=0.5.
n
Parameters
Means
Bias
RMSE
25
α
=
0.5
0.4849
-
0.0151
0.1974
β
=
0.5
0.5379
0.0379
0.2012
γ
=
0.5
0.5469
0.0469
0.1134
θ
=
0.5
0.5552
0.0552
0.3301
50
α
=
0.5
0.4913
-
0.0087
0.1763
β
=
0.5
0.5204
0.0204
0.1655
γ
=
0.5
0.5284
0.0284
0.0861
θ
=
0.5
0.5544
0.0544
0.2889
100
α
=
0.5
0.5080
0.0080
0.1603
β
=
0.5
0.5131
0.0131
0.1334
γ
=
0.5
0.5100
0.0100
0.0658
θ
=
0.5
0.5394
0.0394
0.2355Table 2
Simulation study atα=1.0, β=1.0, γ=1.0, and θ=1.0.
n
Parameters
Means
Bias
RMSE
25
α
=
1.0
0.8897
-
0.1103
0.2947
β
=
1.0
1.0938
0.0938
0.3420
γ
=
1.0
1.1110
0.1110
0.2602
θ
=
1.0
1.0330
0.0330
0.2509
50
α
=
1.0
0.9189
-
0.0811
0.2172
β
=
1.0
1.0592
0.0592
0.2449
γ
=
1.0
1.0545
0.0545
0.1671
θ
=
1.0
1.0397
0.0397
0.2082
100
α
=
1.0
0.9490
-
0.0510
0.1593
β
=
1.0
1.0465
0.0465
0.1733
γ
=
1.0
1.0096
0.0096
0.1137
θ
=
1.0
1.0470
0.0470
0.1514Table 3
Simulation study atα=2.0, β=2.0, γ=2.0, and θ=2.0.
n
Parameters
Means
Bias
RMSE
25
α
=
2.0
1.8415
-
0.1585
0.7105
β
=
2.0
2.1605
0.1605
0.7190
γ
=
2.0
2.2196
0.2196
0.4790
θ
=
2.0
2.0191
0.0191
0.3825
50
α
=
2.0
1.9171
-
0.0829
0.5363
β
=
2.0
2.1175
0.1175
0.5653
γ
=
2.0
2.1155
0.1155
0.3550
θ
=
2.0
2.0055
0.0055
0.3407
100
α
=
2.0
1.9881
-
0.0119
0.3892
β
=
2.0
2.0890
0.0890
0.4110
γ
=
2.0
2.0507
0.0507
0.2777
θ
=
2.0
1.9862
-
0.0138
0.2962Remark 3.
It can be deduced from Tables1, 2, and 3 that the root mean square error (RMSE) reduces for all the selected parameter values as the sample size increases. Also, the bias posed by the estimates is closer to the true parameter values and the absolute bias reduces as the sample size increases. Hence, as sample size increases, the estimates tend towards (or approaches) the true parameter values.


## 4. Application

To demonstrate the potentials of the GoLom distribution, a comparison was made using the GoLom distribution and some other compound distributions like Beta Lomax distribution, Weibull Lomax distribution, and Kumaraswamy Lomax distribution. The following criteria were used to select the distribution with the best fit: Negative Log-Likelihood (−LL) value, Akaike Information Criteria (AIC), Bayesian Information Criteria (BIC), Consistent Akaike Information Criteria (CAIC), and Hannan and Quinn Information Criteria (HQIC). The value for the Kolmogorov Smirnov (KS) statistic, Anderson Darling (A) statistic, and thep value are also provided.The data relating to the strengths of 1.5 cm glass fibres which was obtained by workers at the UK National Physical Laboratory was used. The data has previously been used by Smith and Naylor [17], Bourguinon et al. [18], and Merovci et al. [19]. The observations are as follows:0.55, 0.74, 0.77, 0.81, 0.84, 1.24, 0.93, 1.04, 1.11, 1.13, 1.30, 1.25, 1.27, 1.28, 1.29, 1.48, 1.36, 1.39, 1.42, 1.48, 1.51, 1.49, 1.49, 1.50, 1.50, 1.55, 1.52, 1.53, 1.54, 1.55, 1.61, 1.58, 1.59, 1.60, 1.61, 1.63, 1.61, 1.61, 1.62, 1.62, 1.67, 1.64, 1.66, 1.66, 1.66, 1.70, 1.68, 1.68, 1.69, 1.70, 1.78, 1.73, 1.76, 1.76, 1.77, 1.89, 1.81, 1.82, 1.84, 1.84, 2.00, 2.01, 2.24.The performances of the GoLom distribution with the other competing distributions are shown in Table4.Table 4
Performance rating of the GoLom distribution.
Distributions
Estimates
−LL
AIC
CAIC
BIC
HQIC
Gompertz Lomax
α
^
=
0.0046
14.5027
37.0055
37.6951
45.5780
40.3771
β
^
=
8.1791
γ
^
=
1.5158
θ
^
=
0.5069
Weibull Lomax
α
^
=
6.0947
15.3399
38.6798
39.3695
47.2524
42.0514
β
^
=
0.1069
a
^
=
1.0629
b
^
=
0.0649
Beta Lomax
α
^
=
18.1737
24.4034
56.8068
57.4964
65.3793
60.1784
β
^
=
26.7645
a
^
=
10.8769
b
^
=
0.0329
Kumaraswamy Lomax
α
^
=
9.8352
18.1027
44.2055
44.8951
52.7779
47.5771
β
^
=
45.3107
a
^
=
15.1182
b
^
=
0.0483Remark 4.
The distribution that corresponds to the lowest −LL, AIC, CAIC, BIC, and HQIC is judged to be the best out of the competing distributions. With this regard, the competing distributions can be ranked in the following order (best to the least): Gompertz Lomax distribution, Weibull Lomax distribution, Kumaraswamy Lomax distribution, and Beta Lomax distribution.The values for the Kolmogorov Smirnov statistic, Anderson Darling statistic, and thep value are as shown in Table 5.Table 5
Table of test statistic.
Distributions
KS
A
p value
Gompertz Lomax
0.1542
0.9462
0.0998
Weibull Lomax
0.1517
1.3315
0.1100
Beta Lomax
0.2182
3.1986
0.0049
Kumaraswamy Lomax
0.1854
1.9915
0.0263A plot showing all the competing distributions against the empirical histogram of the observed data is as shown in Figure4.Figure 4
Plot showing the competing distributions with the empirical histogram of the observed data.A plot for the empirical cdf of the competing distributions with the empirical cdf of the observed data is as shown in Figure5.Figure 5
Plot for the empirical cdf of the competing distributions.The plots in Figures4 and 5 affirm the results of the analysis that the Gompertz Lomax distribution is more suitable for the data than the other competing distributions.


## 5. Conclusion

The Gompertz Lomax distribution has been successfully derived; expressions for its basic statistical properties which include the reliability function, hazard function, odds function, reversed hazard function, and quantile, median, and distribution of order statistics have been successfully established. The shape of the distribution could be decreasing or inverted bathtub (depending on the value of the parameters). Meanwhile, the shape of its hazard function could be constant, increasing, or decreasing (depending on the value of the parameters). The simulation study that was conducted shows that the parameters of the Gompertz Lomax distribution are stable; though values for biasedness were generated, these values are small, indicating that the maximum likelihood estimates of the GoLom distribution are not too far from the true parameter values; the absolute bias and the root mean square values also decreases as the sample size increases. An application to a real life data shows that the Gompertz Lomax distribution is a strong and better competitor for the Weibull Lomax distribution, Beta Lomax distribution, and Kumaraswamy Lomax distribution.



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