American Journal of www.biomedgrid.com Biomedical Science & Research ISSN: 2642-1747 --------------------------------------------------------------------------------------------------------------------------------- Research Article Copy Right@ Obubu Maxwell Useful Generalization of the Inverse Lomax Distribution: Statistical Properties and Application to Lifetime Data Obubu Maxwell1, Adeleke Akinrinade Kayode2, Ibeakuzie Precious Onyedikachi1, Chinelo Ijeoma Obi-Okpala1, Echebiri Udochukwu Victor3 1Department of Statistics, Nnamdi Azikiwe University, Nigeria 2Department of Statistics, University of Ilorin, Nigeria 3Department of Statistics, University of Benin, Nigeria *Corresponding author: Obubu Maxwell, Department of Statistics, Nnamdi Azikiwe University, Awka, Nigeria To Cite This Article: Obubu Maxwell, Useful Generalization of the Inverse Lomax Distribution: Statistical Properties and Application to Lifetime Data. Am J Biomed Sci & Res. 2019 - 6(3). AJBSR.MS.ID.001040. DOI: 10.34297/AJBSR.2019.06.001040. Received: November 15, 2019; Published: November 26, 2019 Abstract A four-parameter compound continuous distribution for modeling lifetime data known as the Odd Generalized Exponentiated Inverse Lomax distribution was proposed in this study. Basic Structural properties were derived in minute details. 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 existing distribution. square error (RMSE) were obtained. The proposed model was applied to the strengths of glass fibres dataset and the result compared to other Keywords: Odd generalized exponentiated family, Inverse lomax distribution, Moments, Reliability function, Hazard function, Continuous distribution Introduction verted Lomax distribution if the corresponding probability density Various distributions have been proposed to serve as models for function and cumulative density function are given by [5]: wide applications on data from different real-life situations through −+(1α ) the extension of existing distribution. This has been achieved in αβ β gx( ;αβ , )=+ 1 ; x ≥>0,αβ , 0 (1) various ways. The Lomax distribution also called “Pareto type II” xx2 is a special case of the generalized beta distribution of the second −α kind [1], and can be seen in many application areas, such as actu- β Gx( )=+ 1 ;x ≥> 0,αβ , 0 (2) arial science, economics, biological sciences, engineering, lifetime x and reliability modeling and so on [2]. This heavy-duty distribu- In literature exist several families of distribution and the ref- tion is considered useful as an alternative distribution to survival erences to them are listed in [6-7]. In the course of this study, the problems and life-testing in engineering and survival analysis [3]. Odd Generalized Exponential family of distribution developed by Inverse Lomax distribution is a member of the inverted family of [8] having probability density and cumulative distribution function - given by; ations with a realized non-monotonic failure rate [4]. If a random Gx() a−1 distributions and discovered to be very flexible in analyzing situ −b Gx() − abg() x 1−Gx () b f() x =e1 −> e 1−Gx () ;ab , 0 (3) 2 variable Χ has a Lomax distribution, then 1 has an inverse [1−Gx ( )] Y = Lomax Distribution. Thus, a random variable X is saidX to have an In This work is licensed under Creative Commons Attribution 4.0 License AJBSR.MS.ID.001040. 258 Am J Biomed Sci & Res Copy@ Obubu Maxwell a −α a Gx() β −b 1+ x 1−Gx () −b F( x )= 1−> e ;ab , 0 (4) −α β 11−+ x SOGE− IL ( x )=−− 1 1 e αβ, ,ab ,> 0 (7) 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 The failure rate function/hazard function is given by; −α article is structured as follows: In section 2, we derive the cumu- β a−1 −α 1+ β 1+ x x − − b−α b −α β β −+ 11−+ 11 x lative distribution function, probability density function, reliability β −+α x abαβ (1+− ) (1 ) e 1 e x function, odd function, hazard function, reverse hazard function, hxOGE− IL ()= αβ, ,ab ,> 0 (8) −α a β 1+ and cumulative hazard function of the Odd Generalized Exponenti- x − b −α −α 2 β 11−+ β x ated Inverse Lomax distribution, and present their respective plots xe2 11−+ 11 −− x 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 The reverse hazard function is given by; section 3. In section 4, we determined the order statistics and the −α − β −α a 1 1+ β 1+ x x − − b −α b −α β estimate of the unknown parameter using maximum likelihood es- β 11−+ −+ 11 x β −+α x abαβ (1+− ) (1 ) e 1 e x timation techniques. Application to two different lifetime dataset rOGE− IL ()x = αβ, ,ab ,> 0 (9) and a conclusion is made in section 5. −α a β 1+ x − b −α −α 2 β 11−+ β x xe2 11−+ 11 −− The Odd Generalized Exponentiated Inverse Lomax x (OGE-IL) distribution A four parameter Odd Generalized Exponentiated Inverse Lo- The cumulative hazard function is given by; max distribution was investigated in this section. Obtained by sub- a −α stituting equation (2) into (4), the cumulative distribution function β 1+ x −b of the OGE-IL distribution is given by; −α β 11−+ x =− −− αβ > a HOGE− IL ( x ) log 1 1 e , ,ab , 0 (10) −α β 1+ x −b −α β 11−+ x FOGE− IL ( x )= 1−> e αβ, ,ab , 0 (5) And the odd function is given by; −α a β 1+ x The corresponding probability density function is given by; − b −α β 11−+ x a−1 −α −α 1− e β β 1+ 1+ x x −b −b −+(1α ) −α −α β β β abαβ 1+ 11−+ 11−+ x x x F− () x =−>e 1 e ;αβ , ,ab , 0 (6) O() x = αβ, ,ab ,> 0 (11) OGE IL 2 OGE− IL −α a −α β 2 β 1+ x 11−+ x − x b −α β 11−+ x 11−−e The reliability function is given by; Figure 1: Probability density function of the OGE-IL distribution. American Journal of Biomedical Science & Research 259 Am J Biomed Sci & Res Copy@ Obubu Maxwell Figure 2: Cumulativve distribution function of the OGE-IL distribution. Figure 3: Reliability plot of the OGE-IL distribution. Figure 4: Harzard plot of the OGE-IL distribution. The respective plots of the probability density function, cu- −α − β −α a 1 1+ β −+(1α ) 1+ x x αβ β − − mulative distribution function, reliability function, and the hazard b−α b −α ab 1+ β β 2 −+ 11−+ 11 x xx x functions for different parameter values are represented in Figure limfxOGE− IL ( )= lim e1 −= e 0 xx→→00−α 2 β 1-4 −+ 11 x Structural Properties of the Odd Generalized −α a−1 β −α 1+ β −+(1α ) 1+ x x αβ β − − b−α b −α β β Exponentiated Inverse Lomax (OGE-IL) distribution ab 1+ 11−+ 2 −+ 11 x xx x limfxOGE− IL ( )= lim e1 −= e 0 xx→∞ →∞ −α 2 In this section we derive some of the basic theoretical/struc- β −+ 11 tural properties of the OGE-IL distribution, such as; its asymptotic x behavior, quantile, median, Skewness and kurtosis of the Odd Gen- Thus showing that the Odd Generalized Exponentiated Inverse eralized Exponentiated Inverse Lomax distribution. Lomax distribution is unimodal i.e. it has only one mode Asymptotic Behavior of the OGE-IL Distribution limFx ( )= 1 We can also deduce that for the OGE-IL Distribution, x→∞ We examine the behavior of the OGE-IL distribution in equation i.e. (6) as x→0 and as x→∞ and in equation (10) as i.e. x→∞ American Journal of Biomedical Science & Research 260 Am J Biomed Sci & Res Copy@ Obubu Maxwell −α a −1 β −1 1+ x α − 1 b −α β −+ln 1 u a 11−+ x x = β −1 lim 1 −=e 1 1 x→∞ bu−+ln 1 a The Quantile function of the Odd Generalized Exponentiated Thus, indicating that the OGE-IL distribution has a proper prob- Inverse Lomax distribution is; ability density function. −1 −1 Quantile function and Median 1 α −+ln 1 u a Qu() = β −1 (12) −1 1 Q() uF= () u OGE−− IL OGE IL bu−+ln 1 a −α a β 1+ x −b −α β 11−+ u Uniform(0,1) x Let u = 1− e The median of the Odd Generalize Exponentiated Inverse Lo- max distribution can be gotten by placing u as 0.5 in equation (12) above, we obtain −1 −α − β 1 1+ α x 1 −b −α −+ln 1 0.5 a β 11−+ 1 x median = β −1 a 1 ue=1 − b−+ln 1 0.5 a Skewness and kurtosis −α β According to Kenney and Keeping [9], the Bowely Quartile 1+ − x b−α Skewness is given as; β 11−+ 1 x a −+ 1+=ue q(0.75)2 qq (0.5) S = (0.25) (13) k qq− −α (0.75) (0.25) β 1+ 1 x Moors quantile-based Kurtosis by Moors [10] is given as; ln 1+=−uba −α β 11−+ qqqq(0.875)−+ (0.625) + x K = (0.375) (0.125) (14) u qq− (0.75) (0.25) −α β 1 1+ −+ln 1 u a Moment x = −α b β Irr∞ 11−+ µr =E() Χ=∫0 x fOGE− IL () x dx x −α − β −α a 1 1+ β x 1+ −+(1α ) −b x −α − β β b −α 11−+ β abαβ1+ 11−+ ∞ x I r−2 x x −α µ = ∫0 x e 1−e dx −α r 2 1 β −α β β −ln 1 +uba 1 −+ 1 =−1 + 11−+ x x x −α 11β − +aa =+ −+ ln 1u 1 bu ln 1 x American Journal of Biomedical Science & Research 261 Am J Biomed Sci & Res Copy@ Obubu Maxwell −α β 1+ a−1 x −α −α −+(1α ) −+ bk( 1) −α β β β β 1+ 1+ 1+ 11−+ a−1 x x x Ika−1 ∞ r−2 x −b−b µr = ab αβ ∑ (− 1) ∫0 ( x ) e dx −+(1α ) −α −α k=0 2 β β β k −α 11−+ 11−+ β n abαβ1+ x x x 11−+ Lx( ;αβ , ,,) ab = ∏ e 1−e x i 2 i=1 −α 2 β x 11−+ x Using binomial expansion and solving through, we obtained an expression for the moment as; wk− j |1∞∞a −1 aj−−12 (−+ 1) (k 1) rw+ Thus the log likelihood function becomes; µr =∑∑∑ywk=0 = 00 = βαb aB(1− r , α ( k ++ w y ) + r ) (15) ky w! −α β −α 1+ x β − b −α β 1+ −+ −α 11 β β x x αβ=+++− α β nn − −+ −+α n +− n + −n − Lx(i ; , , abn , ) ln n ln nanb ln ln 2∑∑ii=11 ln xi 2= ln 1 1 (1 )∑i=1 ln 1 b ∑i=1 −α (a 1)∑i=1 ln 1 e (19) x xi β 11−+ Order Statistics x Consider random sample denoted by ΧΧ....