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A Theoretical Analysis of the Odd Generalized Exponentiated Inverse Lomax Distribution

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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 −1 QFOGE−− 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 e1−= e 0 11β 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 e1−= 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) 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. 20 −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 ∑∑∑ ( ) ywk=0 = 00 = ky w! Moment ∞ (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.
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