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The Burr X-Exponential Distribution: Theory and Applications

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The Burr X-Exponential Distribution: Theory and Applications View metadata, citation and similar papers at core.ac.uk brought to you by CORE Proceedings of the World Congress on Engineering 2017 Vol I provided by Covenant University Repository WCE 2017, July 5-7, 2017, London, U.K. The Burr X-Exponential Distribution: Theory and Applications Pelumi E. Oguntunde, Member, IAENG, Adebowale O. Adejumo, Enahoro A. Owoloko, Manoj K. Rastogi, and Oluwole A. Odetunmibi data indicates that the Bur X-Lomax distribution is more Abstract— In this research, the Burr X-Exponential flexible than the Lomax and other competing distributions. distribution was defined and explored using the Burr X There are other generalized families of distributions like the family of distributions. Its basic statistical properties were Beta-G (Eugene at al., 2002), Kumaraswamy-G (Cordeiro and identified and the method of maximum likelihood was de Castro, 2011), Weibull-G (Bourguinon et al., 2014), proposed in estimating the model parameters. The model Weibull-X (Alzaatreh et al., 2013), Transmuted-G (Shaw and was applied to three different real data sets to assess its Buckley, 2007), Logistic-X (Tahir et al., 2016), Marshall- flexibility over its baseline distribution. Olkin-G family of distributions (Marshall and Olkin, 1997) and many others, but of interest to us in this research is the Index Terms— Burr X distribution, Burr X family, Burr X-family of distributions. Exponential distribution, Properties The cdf and pdf of the Burr X family of distribution is given by; 2 I INTRODUCTION Gx Fx 1 exp (3) The Burr distribution has different forms, of these; the Burr- 1Gx Type X and XII distributions have both received appreciable usage in probability distribution theory. Interestingly, the and Burr-Type X distribution is related to some well-known 1 22 standard theoretical distributions like the Weibull distribution 2 g x G x G x G x and Gamma distribution. fx exp 1 exp 3 11G x G x The cdf and pdf of the Burr X distribution are given by; 1Gx 2 F( x ) 1 ex (1) (4) respectively. and for x 0, 0 22 1 f( x ) 2 xexx 1 e (2) where; is a shape parameter whose role is to vary tail weight. respectively Gx and gx are the cdf and pdf of the for x 0, 0 baseline distribution respectively. where; is the scale parameter. This research is aimed at studying and exploring the Burr X- Recently, the Burr X distribution has been used as a generator Exponential distribution using the family of distribution of other compound distributions by Yousof et al., (2016). This defined in (3) and (4) respectively. In the next section, the new family of distribution has been used to extend the densities and properties of the Burr X-Exponential Weibull and Lomax distributions. An application to real life distribution are derived. Manuscript received: February 16, 2017; revised: March 12, 2017. This II THE BURR X-EXPONENTIAL DISTRIBUTION work was supported financially by Covenant University, Ota, Nigeria. Consider a random variable X with a cdf and pdf defined by; P. E. Oguntunde, E. A. Owoloko and O. A. Odetunmibi work with the Department of Mathematics, Covenant University as lecturer (corresponding G x 1 exp x (5) author phone: +234 806 036 9637; e-mail: [email protected]; [email protected], and alfred.owoloko@[email protected], [email protected]). g x exp x (6) A. O. Adejumo works with Department of Statistics, University of Ilorin respectively. Nigeira and Department of Mathematics, Covenant University, Nigeria. ([email protected]; [email protected]). for x 0, 0 M. K. Rastogi works with National Institute of Pharmaceutical Education where; is a scale parameter and Research, Hajipur- 844102, India (e-mail: [email protected]) ISBN: 978-988-14047-4-9 WCE 2017 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering 2017 Vol I WCE 2017, July 5-7, 2017, London, U.K. Then, the cdf of the Burr X-Exponential distribution is where; and fx are as defined in Equations (7) derived by substituting equation (5) into Equation (3) to give; and (8) respectively. 2 1 exp x Therefore; 1 Fx 1 exp x 22 xx expx 21 e 11ee exp 1 exp 2 xx (7) x ee e For x 0, 0, 0 hx 2 x Its corresponding pdf is given by; 1 e x 2 1 1 exp x x 21 e 1 e e fxexp 2 x x e e (12) for 1 2 x 1 e 1 exp x e It is interesting to note that the plots at various parameter values indicate that the shape of the hazard function of the (8) Burr X-Exponential distribution is increasing. for Odds Function where; is a shape parameter Odds function is mathematically defined by: Fx is a scale parameter Ox (13) Sx Therefore, the odds function for the Burr X-Exponential The shape of the Burr X-Exponential distribution could be distribution is: unimodal (for instance, when ) or decreasing (for 2 " 2, 3"," 2, 0.3" 1 exp x instance, when " 0.3, 0.5"," 0.7, 2" ). 1 exp expx Ox 2 Reliability Analysis 1 exp x Here, the survival function, hazard function, odds function 1 1 exp expx and reversed hazard function for the Burr X-Exponential distribution are derived. (14) for Survival Function The mathematical expression for survival function is; Reversed Hazard Function S x 1 F x (9) Reversed hazard function is given by: Where; Fx is as defined in Equation (7). fx rx (15) Therefore, the expression for the survival function of the Burr Fx X-Exponential distribution is: Therefore, the expression for the reversed hazard function of 2 the Burr X-Exponential distribution is: 1 exp x Sx 1 1 exp expx 2 2 1 exp x 1 exp x 2 exp (10) expx expx for rx 2 1 exp x Hazard Function 1 exp expx The mathematical expression for hazard function is: fx hx (11) (16) 1 Fx for ISBN: 978-988-14047-4-9 WCE 2017 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering 2017 Vol I WCE 2017, July 5-7, 2017, London, U.K. Quantile Function and Median The quantile function is derived from; 1 1 Q u F u (17) 1 X log 1 1 2 Therefore, the quantile function for the Burr X-Exponential log 1 u 1 distribution is given by; (20) where; . 1 1 Qu log 1 1 2 log 1 u 1 Estimation of Parameters Let x, x ,..., x denote random samples from the Burr X- (18) 12 n Exponential distribution with parameters and , using the where; . u Uniform0,1 method of maximum likelihood estimation (MLE), the That means, random samples can be generated from the Burr likelihood function is given by: X-Exponential distribution using: 1 22 n 2 1 exp x 1 exp xx 1 exp i ii f x12, x ,..., xn ; , 2 exp 1 exp expxx exp i1 expxi ii Let l log f x12 , x ,..., xn ; , denote the log-likelihood function, then: 2 n n n 1 exp x l nlog 2 n log n log log 1 exp x 2 x i ii i1 i 1 i 1 expxi 2 n 1 exp x i 1 log 1 exp i1 expxi (21) The summary of Data I is shown in Table 1: Differentiating Equation (21) with respect to parameters and , setting the resulting non-linear system of equations to Table 1: Summary of Data on Height of 100 Female Athletes zero and solving them simultaneously gives the maximum N Mean Variance Skewness Kurtosis likelihood estimates of parameters and respectively. It 100 174.6 67.9339 -0.5598 4.1967 is much easier to solve these equations using algorithms in statistical software like R and so on when data sets are The performance of the Burr X-Exponential distribution with available. respect to its baseline distribution is as shown in Table 2: Table 2: Burr X-Exponential distribution Versus Exponential III APPLICATIONS TO REAL LIFE DATA distribution (with standard error in parentheses) Distributions Estimates Log- AIC In this section, the Burr X-Exponential distribution and Likelihood Exponential distribution are applied to three real data. Here, Burr X- -747.0396 1498.079 judgment is based on the Log-likelihood and Akaike Exponential 1.563e 01 Information Criterion (AIC) values posed by these 1.696e 02 distributions. Data I: It represents the height of 100 female athletes 2.292e 04 (measured in cm) collected at the Australian Institute of Sport. 3.041e 05 The data has previously been used by Cook and Weisberg (1994), Al-Aqtash et al., (2014) and Owoloko et al., (2016). Exponential -616.2463 1234.493 0.0057276 0.0005729 ISBN: 978-988-14047-4-9 WCE 2017 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) Proceedings of the World Congress on Engineering 2017 Vol I WCE 2017, July 5-7, 2017, London, U.K. DATA II: The second data is from an accelerated life test of Remarks: The lower the AIC value, the better the model and 59 conductors. The data has previously been used by Nasiri et the higher the log-likelihood value the better the model. al., (2011) and Oguntunde et al., (2016). The observations are as follow: IV CONCLUSION The data summary is as shown in Table 3: The Burr X-Exponential distribution has been successfully developed, its statistical properties like the quantile function, Table 3: Summary of data on accelerated life test of median, survival function, hazard function, reversed hazard conductors function and odds function have been explicitly established.
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