A Theoretical Analysis of the Odd Generalized Exponentiated Inverse Lomax Distribution

Biometrics & Biostatistics International Journal

Research Article Open Access A theoretical analysis of the odd generalized exponentiated inverse Lomax distribution

Abstract Volume 8 Issue 1 - 2019 A four parameter compound continuous distribution for modeling lifetime data known Obubu Maxwell,1 Agu Friday,I,2 Nwokike as the Odd Generalized Exponentiated Inverse Lomax distribution was proposed in 3 4 this study. Basic Structural properties were derived in minute details. Chukwudike C, Francis Runyi E, Offorha Bright C3 Keywords: odd generalized Exponentiated family, inverse lomax distribution, 1Department of Statistics, Nnamdi Azikiwe University, Nigeria moments, reliability function, hazard function, continuous distribution 2Department of Statistics, University of Calabar, Nigeria 3Department of Statistics, Abia State University, Nigeria 4Department of Mathematics and Computer Science, Arthur Jarvis University, Nigeria

Correspondence: Obubu Maxwell, Department of Statistics, Nnamdi Azikiwe University, Awka, Nigeria, Email

Received: January 17, 2019 | Published: February 11, 2019

Introduction and tractable than the Inverse Lomax distribution. The rest of the article is structured as follows: In section 2, we derive the cumulative Various distributions have been proposed to serve as models for distribution function, probability density function, reliability function, wide applications on data from different real-life situations through odd function, hazard function, reverse hazard function, and cumulative the extension of existing distribution. This has been achieved in hazard function of the Odd Generalized Exponentiated Inverse various ways. The Lomax distribution also called “Pareto type II” is Lomax distribution, and present their respective plots for different 1 a special case of the generalized beta distribution of the second kind, values of the parameter. In minute details, we establish the structural and can be seen in many application areas, such as actuarial science, properties, which include the asymptotic behavior, moments, quantile economics, biological sciences, engineering, lifetime and reliability function, median, Skewness and kurtosis, in section 3. In section 4, 2 modeling and so on. This heavy duty distribution is considered useful we determined the order statistics and the estimate of the unknown as an alternative distribution to survival problems and life-testing in parameter using maximum likelihood estimation techniques. 3 engineering and survival analysis. Inverse Lomax distribution is a Application to two different lifetime dataset and a conclusion is made member of the inverted family of distributions and discovered to be in section 5. very flexible in analyzing situations with a realized non-monotonic failure rate.4 If a random variable X has a Lomax distribution, then 1 The Odd Generalized Exponentiated Inverse Y = has an inverse Lomax Distribution. Thus, a random variable X Lomax (OGE-IL) distribution X is said to have an Inverted Lomax distribution if the corresponding probability density function and cumulative density function are A four parameter Odd Generalized Exponentiated Inverse Lomax given by Yadav et al.5 distribution was investigated in this section. Obtained by substituting −+(1 α ) αβ β equation (2) into (4), the cumulative distribution function of the OGE- gx( ;αβ ,) =+ 1 ; x ≥> 0,αβ , 0 (1) x2 x IL distribution is given by;  a −α −α β β 1+ x Gx( ) =+≥> 1 ; x 0, αβ , 0 (2) −b  −α x β 11−+ In literature exist several family of distribution and the references =−>x αβ 6,7 FOGE− IL ( x) 1 e , ,,ab 0 (5) to them are listed in Alizadeh et al. In the course of this study, the  Odd Generalized Exponential family of distribution developed by  Alizadeh8 having probability density and cumulative distribution   function given by; 

a−1 The corresponding probability density function is given by; Gx( )  Gx( ) −bb  −  α −1 abg( x) 11−−Gx( )  Gx( ) −−αα   ââ f( x) =e 1 −>e ;, ab 0(3) 11++ 2  −+1 α xx ( ) −−bb 1− Gx β −−αα ( )  abαβ 1+ ââ   11−+ 11−+ x xx a fOGE− IL ( x) =e 1 −>e ; αβ , ,,ab 0 Gx( ) −α 2 −  β  b 2 1−Gx( ) x 11−+  F( x) =1−> e;, ab 0 (4) x    (6) was used to extend the Inverse Lomax distribution. The goal is The reliability function is given by; to derive a continuous compound distribution that would be robust

Submit Manuscript | http://medcraveonline.com Biom Biostat Int J. 2019;8(1):17‒22. 17 ©2019 Maxwell et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and build upon your work non-commercially. Copyright: A theoretical analysis of the odd generalized exponentiated inverse Lomax distribution ©2019 Maxwell et al. 18

a −α β The respective plots of the probability density function, cumulative 1+ x −b distribution function, reliability function, and the hazard functions for −α β different parameter values are represented in Figure 1–4. 11−+ x SOGE− IL ( x) =−−1 1 e αβ , ,,ab> 0 (7)      The failure rate function/hazard function is given by;

a−1 −−αα   ββ   11++ xx   −−bb −−αα   −+α â β  (1 ) 11−+ 11−+  β xx   abαβ 11+−e e x     hx= αβ, ,,ab> 0 OGE− IL ( ) a −α β 1+ x −b −α −α 2  β  β 11−+ xe2  11−+  11 −−  x  (8)  x      Figure 1 Probability density function of the OGE-IL distribution.

The reverse hazard function is given by;

a−1 −−αα ββ 11++ xx −−bb −−αα −+α ββ (1 ) 11−+ 11−+ β xx abáeβ 11+− e x     rx= α,,,âab> 0 OGE− IL ( ) a −α â 1+ x  −b (9) −α −α 2 â â 11−+  xe2 11−+ 1 −  x   x       The cumulative hazard function is given by; a −α β 1+ x Figure 2 Cumulativve distribution function of the OGE-IL distribution. −b −α β 11−+ x HOGE− IL ( x) =−log 1 −− 1 e αβ , ,ab ,> 0 (10)       And the odd function is given by;

a −α β 1+ x −b −α β 11−+ 1− e x    (11)  = αβ > OOGE− IL ( x) a , ,,ab 0 −α β 1+ x −b −α β 11−+ 11−−e x    Figure 3 Reliability plot of the OGE-IL distribution. 

Citation: Maxwell O, Friday AI, Chukwudike NC, et al. A theoretical analysis of the odd generalized exponentiated inverse Lomax distribution. Biom Biostat Int J. 2019;8(1):17‒22. DOI: 10.15406/bbij.2019.08.00264 Copyright: A theoretical analysis of the odd generalized exponentiated inverse Lomax distribution ©2019 Maxwell et al. 19

Quantile function and Median − QF() = 1 () OGE−− IL OGE IL a −α β 1+ x −b −α β 11−+ Let u=1 − e x      −α β 1+ x −b −α β 1 11−+ uea =1 − x     Figure 4 Harzard plot of the OGE-IL distribution. −α β 1+ x −b Structural Properties of the Odd Generalized −α β 1 11−+ Exponentiated Inverse Lomax (OGE-IL) x 1+=uea  distribution −α β 1+ In this section we derive some of the basic theoretical/structural 1  a x properties of the OGE-IL distribution, such as; its asymptotic behavior, ln 1+=−ub −α  β quantile, median, Skewness and kurtosis of the Odd Generalized 11−+ x Exponentiated Inverse Lomax distribution. 

Asymptotic Behavior of the OGE-IL Distribution −α 1 β −+ln 1 u a 1+ We examine the behavior of the OGE-IL distribution in equation x  =  (6) as x → ∞ and as x → ∞ and in equation (10) as i.e. x → ∞ −α b β a−1  −−αα 11−+ ββ  11++ x −+(1 α ) xx  −−bb αβ β −−αα ab 1+ ββ 2  11−+ 11−+ x x xx −α lim fxOGE− IL ( ) =lim e1−= e 0 11β  xx→→00−α 2 aa β  −+ln 1u =+ 1  bu −+ln 1 11−+  x  x   −1  −1 α 1 a−1 a −−αα −+ln 1 u ββ  11++  −+(1 α ) xx  −−bb x = β −1 αβ β −−αα ab 1+ ββ 1 2  11−+ 11−+ −+a x x xx buln 1 lim fxOGE− IL ( ) =lim e1−= e 0  xx→→∞∞−α 2  β   −+  11 The Quantile function of the Odd Generalized Exponentiated x  Inverse Lomax distribution is; −1 −1 α Thus showing that the Odd Generalized Exponentiated Inverse 1 −+ln 1 u a Lomax distribution is unimodal i.e. it has only one mode.  Qu( ) = β −1 (12) 1 We can also deduce that for the OGE-IL Distribution, limFx( ) = 1 a x→∞ bu−+ln 1 a  i.e. −α  β  1+ x −b u ~ Uniform(0,1) . −α β 11−+ The median of the Odd Generalize Exponentiated Inverse Lomax lim 1−=e x 1 x→∞  distribution can be gotten by placing in equation (12) above,  we obtain −1 −1  α  1  −+ln 1( 0.5) a  Thus indicating that the OGE-IL distribution has a proper median = β −1 1 a probability density function. b −+ln 1( 0.5)  

−α Skewness and Kurtosis β 1+ −+(1 α ) x −+bk( 1) 9 β −α According to Kenney & Keeping, the Bowely Quartile Skewness 1+ β a−1 ∞  11−+ 1 a −1 k r−2 x x is given as; µ=abαβ (−1) x e dx q−+2 qq r ∑ ∫ ( ) 2 (0.75) ( 0.5) ( 0.25) = k −α S = (13) k 0 0 β k − 11−+ qq(0.75) ( 0.25) x

Moors quantile based Kurtosis by Moors [10] is given as; Using binomial expansion and solving through, we obtained an qqqq−−+ expression for the moment as; (0.875) ( 0.625) ( 0.375) ( 0.125) wk+ j K = (14) ∞∞a−1 aj−−12  (−+11) (k ) u − µ=1 βαrwáb1+ aB1,− r( k + w + y) + r qq(0.75) ( 0.25) r ∑∑∑   ( ) = = = ky   w! Moment ywk0 00 ∞ (15) µ=1 rr = r E( X) ∫ x fOGE− IL ( x) dx Order statistics 0 a−1 … −−ααConsider random sample denoted by XX1 n from the densities ββ 11++ of the Odd Generalized Exponentiated Inverse Lomax distribution. −+1 α xx ( ) −−bb β −−αα abαβ 1+ ββ Then, ∞  11−+ 11−+ 1 r−2 x xx n! j−1 nj− µ=r x e 1−e dx fx= fxFx1− Fx ∫ ( ) −α 2 j: n ( ) OGE−− IL( ) OGE IL ( ) OGE − IL ( ) 0   ( jn−−1!) ( 1!) β 11−+  th x The pdf of the j order statistics for the OGE-IL distribution is given as;

j−−1 nj a−1 aa −−αα −−αα  ββ  β β 11++  1 +1+ −+1 α xx xx ( ) −−bb − b −b β −−αα−−αα abαβ+1 ββ ββ  11−+ 11−+ 11−+ 11−+ n! x xx xx =  −−  −−  (16) fxjn: ( ) 2 e 11e e 11e ( j−−1!) ( nj) !−α  2 β  x 11−+  x        The minimum order statistics for the OGE-IL distribution is given as;

n−1 a−1 a −−α á −α ββ β 11++ 1+ −+1 α xx x ( ) −−bb −b β −−αα−α abαβ+1 ββ β  11−+ 11−+ 11−+ x xx x =  −  −−  fxn1:n ( ) 2 e 1e 11e (17) −α  2 β  x 11−+  x      And the maximum order statistics for the OGE-IL distribution is given as; n−1 a−1 a −−αα − α ββ  β 11++  1 + −+1 α xx  x ( ) −−bb − b β −−αα−α abαβ 1+ ββ β  11−+ 11−+ 11−+ x xx x =  −−  fxnnn: ( ) 2 e 11e e (18) −α  2 β  x 11−+  x      Parameter estimation

Using maximum likelihood estimation techniques, Let XX1,… n indicate a random sample of the complete OGE-ILD data, and then the sample’s likelihood function is given as; a−1 n −−αα ββ L= f( x;,αβ ,, ab) 11++ Π i −+(1 α ) xx = −−bb i 1 β −−αα abαβ 1+ ββ n  11−+ 11−+ x xx αβ =   −  L( xi ;, ,, ab) Π 2 e 1 e i=1 −α  2 β x 11−+  x  

Thus the log likelihood function becomes;  −α   β  1+ −α  x   −b β  −α  −α 1+  β  nn n n n11−+ (19) ββ x x αβ=+++− α β α − −+ −+α +− + − −   Lxabnnnnbx( ii; , , ,) ln ln ln ln 2∑∑ ln 2 ln 1 1 (1)∑ ln 1 b ∑−α (a1) ∑ ln 1 e xx ii=11=  i=1i i= 1β i=1 11−+  x    By taking the derivative with respect to á, âab ,, , and equating to zero, we have; −α β 1+ x −b −α −α −−ααβ 11−+ β ββx nn1+21α  ++n1 e ∂ln Lx( ) n x−1  xx = −bα −−(11αβ) (x +) + +baα( −) = 0 ∑∑−α 2 −α ∑−α ∂ββii=11 = i−1β β β 1+ (x ++−β )11    x (20) 11+ −+(x β )  −b x x −α   −α 2 β 11−+ β x (x ++−β )111− e x       −α   â  1+  x  −b  −α  −−αα −αâ 11−+ ââ  ââ ââ x  1++ ln 1 1 ++ ln 1 ln 1++ 1 e ∂ln Lx( ) nâ nxx n  xx nn xx =−2 −α +b −ln ln 1 ++ba( −1) = 0 ∑ ∑−α 2 ∑∑ −α ∂âxα i=1â i= 1   ii= 11 − â â 1+ 11−+ x 11+−+(xâ ) −b x x  −α (21)  −α 2 â   11−+  â x 1+ −− 11  e  x      −α  â 1+ x −b −α â ∂ln Lx( ) n n 11−+ =+−∑ln 1 e x =0 (22) ∂a α i=1      −α â 1+ −α x −b â −α 1+ â −α  11−+ â x x 1+ −be x −α −b −α â â −+ −+ 11 ∂ln Lx( ) n nn11 x =− − x +−  = (23) ∑∑11ea( ) −α 0 ∂bbâ ii=11= 1+ x  −b −α   â  11−+ 1− e  x     The parameters of the OGE-IL distribution is obtained by solving equation 20-24. Conclusion Funding Using the Odd Generalized Exponential family of distribution, we The study received no formal support. have successfully extended the Inverse Lomax distribution from a two parameter to a four parameter distribution. Densities and theoretical Conflicts of interest characteristics for the proposed OGE-IL distribution have been The Authors declare that they are no competing interest. derived. References Acknowledgments 1. Kleiber C, Kotz S. Statistical size distributions in economics and None. actuarial sciences. New Jersey: John Wiley & Sons; 2003.

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