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.
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 [
Modified and extended versions of the Lomax distribution have been studied; examples include the weighted Lomax distribution [
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. [
The rest of this article is therefore organized as follows: in Section
To start with, the cumulative distribution function (cdf) and probability density function (pdf) of the Lomax distribution with parameters
According to Alizadeh et al. [
Now, if the density in (
Plots for the pdf of the GoLom distribution at various selected values are displayed in Figure
Plot for the pdf of GoLom distribution.
It is clear in Figure
Following Alizadeh et al. [
The associated pdf can be expressed as a linear mixture of the exponentiated Lomax function as follows:
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.
Survival function of GoLom distribution at
Plot for the hazard function of GoLom distribution.
It can be deduced from Figure
Quantile function can be derived from
Random numbers can be generated from the GoLom distribution using
Let
The parameters of the GoLom distribution can be estimated using the method of maximum likelihood (MLE) as follows: let
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 number
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 Tables
Simulation study at
|
Parameters | Means | Bias | RMSE |
---|---|---|---|---|
25 |
|
0.4849 |
|
0.1974 |
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0.5379 | 0.0379 | 0.2012 | |
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0.5469 | 0.0469 | 0.1134 | |
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0.5552 | 0.0552 | 0.3301 | |
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50 |
|
0.4913 |
|
0.1763 |
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0.5204 | 0.0204 | 0.1655 | |
|
0.5284 | 0.0284 | 0.0861 | |
|
0.5544 | 0.0544 | 0.2889 | |
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100 |
|
0.5080 | 0.0080 | 0.1603 |
|
0.5131 | 0.0131 | 0.1334 | |
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0.5100 | 0.0100 | 0.0658 | |
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0.5394 | 0.0394 | 0.2355 |
Simulation study at
|
Parameters | Means | Bias | RMSE |
---|---|---|---|---|
25 |
|
0.8897 |
|
0.2947 |
|
1.0938 | 0.0938 | 0.3420 | |
|
1.1110 | 0.1110 | 0.2602 | |
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1.0330 | 0.0330 | 0.2509 | |
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50 |
|
0.9189 |
|
0.2172 |
|
1.0592 | 0.0592 | 0.2449 | |
|
1.0545 | 0.0545 | 0.1671 | |
|
1.0397 | 0.0397 | 0.2082 | |
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100 |
|
0.9490 |
|
0.1593 |
|
1.0465 | 0.0465 | 0.1733 | |
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1.0096 | 0.0096 | 0.1137 | |
|
1.0470 | 0.0470 | 0.1514 |
Simulation study at
|
Parameters | Means | Bias | RMSE |
---|---|---|---|---|
25 |
|
1.8415 |
|
0.7105 |
|
2.1605 | 0.1605 | 0.7190 | |
|
2.2196 | 0.2196 | 0.4790 | |
|
2.0191 | 0.0191 | 0.3825 | |
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50 |
|
1.9171 |
|
0.5363 |
|
2.1175 | 0.1175 | 0.5653 | |
|
2.1155 | 0.1155 | 0.3550 | |
|
2.0055 | 0.0055 | 0.3407 | |
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100 |
|
1.9881 |
|
0.3892 |
|
2.0890 | 0.0890 | 0.4110 | |
|
2.0507 | 0.0507 | 0.2777 | |
|
1.9862 |
|
0.2962 |
It can be deduced from Tables
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 the
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 [
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 Table
Performance rating of the GoLom distribution.
Distributions | Estimates | −LL | AIC | CAIC | BIC | HQIC |
---|---|---|---|---|---|---|
Gompertz Lomax |
|
14.5027 | 37.0055 | 37.6951 | 45.5780 | 40.3771 |
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Weibull Lomax |
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15.3399 | 38.6798 | 39.3695 | 47.2524 | 42.0514 |
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Beta Lomax |
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24.4034 | 56.8068 | 57.4964 | 65.3793 | 60.1784 |
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Kumaraswamy Lomax |
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18.1027 | 44.2055 | 44.8951 | 52.7779 | 47.5771 |
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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 the
Table of test statistic.
Distributions | KS |
|
|
---|---|---|---|
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 | 0.0263 |
A plot showing all the competing distributions against the empirical histogram of the observed data is as shown in Figure
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 Figure
Plot for the empirical cdf of the competing distributions.
The plots in Figures
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.
The authors declare that there are no conflicts of interest.
The authors are grateful to Covenant University for providing funding and an enabling environment for this research.