We propose a four parameter compound continuous distribution in this study. Simulation studies was carried out to investigate the behavior of the proposed distribution, from which the maximum likelihood estimates for the true parameters, including the bias and root mean square error (RMSE) were obtained. The proposed model was applied to the strengths of glass fibres dataset and the result compared to other existing distribution.

Keywords: odd generalized Exponentiated family, inverse lomax distribution, moments, reliability function, hazard function, continuous distribution

$g(x;\alpha ,\beta )=\frac{{\displaystyle \alpha \beta }}{{\displaystyle x^2}}{(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-(1+\alpha )}; x\ge 0, \alpha ,\beta >0$   (1)

$G\left(x\right)= {(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha } ; x\ge 0, \alpha ,\beta >0$   (2)

$f\left(x\right)=\frac{{\displaystyle abg\left(x\right)}}{{\displaystyle {[1-G\left(x\right)]}^2}}{e}^{-b\left(\frac{{\displaystyle G\left(x\right)}}{{\displaystyle 1-G\left(x\right)}}\right)}{[1-e^{-b\left(\frac{{\displaystyle G\left(x\right)}}{{\displaystyle 1-G\left(x\right)}}\right)}]}^{a-1}; a,b>0$   (3)

$f\left(x\right)=\frac{{\displaystyle abg\left(x\right)}}{{\displaystyle {[1-G\left(x\right)]}^2}}{e}^{-b\left(\frac{{\displaystyle G\left(x\right)}}{{\displaystyle 1-G\left(x\right)}}\right)}{[1-e^{-b\left(\frac{{\displaystyle G\left(x\right)}}{{\displaystyle 1-G\left(x\right)}}\right)}]}^{a-1}; a,b>0$   (4)

was used to extend the Inverse Lomax distribution. The goal is to derive a continuous compound distribution that would be robust and tractable than the Inverse Lomax distribution. The rest of the article is structured as follows: In section 2, we derive the cumulative distribution function, probability density function, reliability function, odd function, hazard function, reverse hazard function, and cumulative hazard function of the Odd Generalized Exponentiated Inverse Lomax distribution, and present their respective plots for different values of the parameter. In minute details, we establish the structural properties, which include the asymptotic behavior, moments, quantile function, median, Skewness and kurtosis, in section 3. In section 4, we determined the order statistics and the estimate of the unknown parameter using maximum likelihood estimation techniques. Application to two different lifetime dataset and a conclusion is made in section 5.

$F_{OGE-IL}\left(x\right)={[1-e^{-b\left(\frac{{\displaystyle {(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}{{\displaystyle 1-{(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}\right)}]}^a \alpha ,\beta ,a,b>0$   (5)

$f_{OGE-IL}\left(x\right)=\frac{{\displaystyle ab\alpha \beta {(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-(1+\alpha )}}}{{\displaystyle x^2{[1-{(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }]}^2}}{e}^{-b\left(\frac{{\displaystyle {(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}{{\displaystyle 1-{(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}\right)}{[1-e^{-b\left(\frac{{\displaystyle {(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}{{\displaystyle 1-{(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}\right)}]}^{a-1}; \alpha ,\beta ,a,b>0$   (6)

$S_{OGE-IL}\left(x\right)=1-{[1-e^{-b\left(\frac{{\displaystyle {(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}{{\displaystyle 1-{(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}\right)}]}^a \alpha ,\beta ,a,b>0$   (7)

$h_{OGE-IL}\left(x\right)=\frac{{\displaystyle ab\alpha \beta {(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-(1+\alpha )}{e}^{-b\left(\frac{{\displaystyle {(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}{{\displaystyle 1-{(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}\right)}{[1-e^{-b\left(\frac{{\displaystyle {(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}{{\displaystyle 1-{(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}\right)}]}^{a-1}}}{{\displaystyle x^2\left({[1-{(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }]}^2\right)(1-{[1-e^{-b\left(\frac{{\displaystyle {(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}{{\displaystyle 1-{(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}\right)}]}^a)}}\alpha ,\beta ,a,b>0$   (8)

$r_{OGE-IL}\left(x\right)=\frac{ab\alpha \beta {\left(1+\frac{\beta }{x}\right)}^{-\left(1+\alpha \right)}{e}^{-b\left(\frac{{\left(1+\frac{\beta }{x}\right)}^{-\alpha }}{1-{\left(1+\frac{\beta }{x}\right)}^{-\alpha }}\right)}{\left[1-e^{-b\left(\frac{{\left(1+\frac{\beta }{x}\right)}^{-\alpha }}{1-{\left(1+\frac{\beta }{x}\right)}^{-\alpha }}\right)}\right]}^{a-1}}{x^2\left({\left[1-{\left(1+\frac{\beta }{x}\right)}^{-\alpha }\right]}^2\right)\left({\left[1-e^{-b\left(\frac{{\left(1+\frac{\beta }{x}\right)}^{-\alpha }}{1-{\left(1+\frac{\beta }{x}\right)}^{-\alpha }}\right)}\right]}^a\right)}\alpha ,\beta ,a,b>0$    (9)

$h_{OGE-IL}\left(x\right)=\frac{{\displaystyle ab\alpha \beta {(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-(1+\alpha )}{e}^{-b\left(\frac{{\displaystyle {(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}{{\displaystyle 1-{(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}\right)}{[1-e^{-b\left(\frac{{\displaystyle {(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}{{\displaystyle 1-{(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}\right)}]}^{a-1}}}{{\displaystyle x^2\left({[1-{(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }]}^2\right)(1-{[1-e^{-b\left(\frac{{\displaystyle {(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}{{\displaystyle 1-{(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}\right)}]}^a)}}\alpha ,\beta ,a,b>0$   (10)

$O_{OGE-IL}\left(x\right)=\frac{{\displaystyle {[1-e^{-b\left(\frac{{\displaystyle {(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}{{\displaystyle 1-{(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}\right)}]}^a}}{{\displaystyle 1-{[1-e^{-b\left(\frac{{\displaystyle {(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}{{\displaystyle 1-{(1+\frac{{\displaystyle \beta }}{{\displaystyle x}})}^{-\alpha }}}\right)}]}^a}} \alpha ,\beta ,a,b>0$   (11)

In simulation studies using R software, the behavior of the OGE–IL distribution parameters was investigated. Data sets from the OGE–IL distribution were generated with replication number m=1000, random samples were selected in sizes 50, 100, 200, and 300. The simulation was performed with different true parameter values for four different cases. The selected true parameter values are, $\alpha =0.5,\beta =0.5, a=0.5, and b=0.5$ ; $\alpha =1.0,\beta =1.0, a=1.0, and b=1.0$ ; $\alpha =2.0,\beta =2.0, a=2.0, and b=2.0$ ; $\alpha =3.0,\beta =3.0, a=3.0, and b=3.0$ for the first, second, third, and fourth cases, respectively. Maximum likelihood estimates for the true parameters, including the Bias and Root Mean Square Error (RMSE) were obtained. The results of the simulation study are displayed in Table 1–4 below. Tables 1–4 clearly indicate that RMSE decreases when the sample size increases for all selected parameter values. In addition, the variations of the predictions are closer to the actual parameter values, with the increase in the sample size decreasing overall bias. Thus, the estimates tend approaches the true parameter value, as the sample size increases.

Data I: strengths of glass fibres dataset

The dataset obtained from Smith & Naylor¹¹ represents the strengths of 1.5 cm glass fibres, measured at the National Physical Laboratory, England. The observations are as follows (Table 5&6);

0.55, 0.93, 1.25, 1.36, 1.49, 1.52, 1.58, 1.61, 1.64, 1.68, 1.73, 1.81, 2.00, 0.74, 1.04, 1.27, 1.39, 1.49, 1.53, 1.59, 1.61, 1.66, 1.68, 1.76, 1.82, 2.01, 0.77, 1.11, 1.28, 1.42, 1.50, 1.54, 1.60, 1.62, 1.66, 1.69, 1.76, 1.84, 2.24, 0.81, 1.13, 1.29, 1.48, 1.50, 1.55, 1.61, 1.62, 1.66, 1.70, 1.77, 1.84, 0.84, 1.24, 1.30, 1.48, 1.51, 1.55, 1.61, 1.63, 1.67, 1.70, 1.78, 1.89.

None.

Author declares that there is no conflict of interest.

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