The Burr X-Exponential Distribution: Theory and Applications

Transcription

1 Proceedings of the World Congress on Engineering 7 Vol I WCE 7, July 5-7, 7, London, U.K. The Burr X- Distribution: Theory Applications Pelumi E. Oguntunde, Member, IAENG, Adebowale O. Adejumo, Enahoro A. Owoloko, Manoj K. Rastogi, Oluwole A. Odetunmibi Abstract In this research, the Burr X- distribution was defined explored using the Burr X family of distributions. Its basic statistical properties were identified the method of maximum likelihood was proposed in estimating the model parameters. The model was applied to three different real data sets to assess its flexibility over its baseline distribution. Index Terms Burr X distribution, Burr X family, distribution, Properties I INTRODUCTION The Burr distribution has different forms, of these; the Burr- Type X XII distributions have both received appreciable usage in probability distribution theory. Interestingly, the Burr-Type X distribution is related to some well-known stard theoretical distributions like the Weibull distribution Gamma distribution. The cdf pdf of the Burr X distribution are given by; F( x) e x x x f ( x) xe e respectively for x, where; is the scale parameter. Recently, the Burr X distribution has been used as a generator of other compound distributions by Yousof et al., (6). This new family of distribution has been used to extend the Weibull Lomax distributions. An application to real life () () data indicates that the Bur X-Lomax distribution is more flexible than the Lomax other competing distributions. There are other generalized families of distributions like the Beta-G (Eugene at al., ), Kumaraswamy-G (Cordeiro de Castro, ), Weibull-G (Bourguinon et al., 4), Weibull-X (Alzaatreh et al., 3), Transmuted-G (Shaw Buckley, 7), Logistic-X (Tahir et al., 6), Marshall- Olkin-G family of distributions (Marshall Olkin, 997) many others, but of interest to us in this research is the Burr X-family of distributions. The cdf pdf of the Burr X family of distribution is given by; F x f x Gx exp Gx (3) g xgx Gx Gx exp exp 3 Gx Gx Gx (4) respectively. for x, where; is a shape parameter whose role is to vary tail weight. G x gx are the cdf pdf of the baseline distribution respectively. This research is aimed at studying exploring the Burr X- distribution using the family of distribution defined in (3) (4) respectively. In the next section, the densities properties of the Burr X- distribution are derived. Manuscript received: February 6, 7; revised: March, 7. This work was supported financially by Covenant University, Ota, Nigeria. P. E. Oguntunde, E. A. Owoloko O. A. Odetunmibi work with the Department of Mathematics, Covenant University as lecturer (corresponding author phone: ; pelumi.oguntunde@covenantuniversity.edu.ng; peluemman@yahoo.com, alfred.owoloko@covenantuniversity@edu.ng, oluwole.odetunmibi@covenantuniversity.edu.ng). A. O. Adejumo works with Department of Statistics, University of Ilorin Nigeira Department of Mathematics, Covenant University, Nigeria. (adejumo.adebowale@covenantuniversity.edu.ng; aodejumo@gmail.com). M. K. Rastogi works with National Institute of Pharmaceutical Education Research, Hajipur- 844, India ( manojlogin@gmail.com) ISBN: ISSN: (Print); ISSN: (Online) II THE BURR X-EXPONENTIAL DISTRIBUTION Consider a rom variable X with a cdf pdf defined by; exp x G x (5) exp x g x (6) respectively. for x, where; is a scale parameter WCE 7

2 Proceedings of the World Congress on Engineering 7 Vol I WCE 7, July 5-7, 7, London, U.K. Then, the cdf of the Burr X- distribution is derived by substituting equation (5) into Equation (3) to give; exp x Fx exp exp x (7) For x,, Its corresponding pdf is given by; x e x e f x exp x x e e x e exp x e for x,, where; is a shape parameter is a scale parameter The shape of the Burr X- distribution could be unimodal (for instance, when ", 3",",.3" ) or decreasing (for instance, when ".3,.5",".7, " ). Reliability Analysis Here, the survival function, hazard function, odds function reversed hazard function for the Burr X- distribution are derived. (8) Survival Function The mathematical expression for survival function is; Where; Fx is as defined in Equation (7). F x (9) Therefore, the expression for the survival function of the Burr X- distribution is: exp x exp exp x for x,, () Hazard Function The mathematical expression for hazard function is: f x h x () F x where; F x f x are as defined in Equations (7) (8) respectively. Therefore; x e x x e e exp exp x x x h x e for x,, e e x e 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 Burr X- distribution is increasing. Odds Function Odds function is mathematically defined by: Fx O x (3) Therefore, the odds function for the Burr X- distribution is: O x exp x exp expx exp x exp expx for x,, (4) Reversed Hazard Function Reversed hazard function is given by: f x r x (5) F x Therefore, the expression for the reversed hazard function of the Burr X- distribution is: r x x exp x exp x exp exp expx exp x exp expx for x,, (6) ISBN: ISSN: (Print); ISSN: (Online) WCE 7

3 Proceedings of the World Congress on Engineering 7 Vol I WCE 7, July 5-7, 7, London, U.K. Quantile Function Median The quantile function is derived from; Q u F u (7) Therefore, the quantile function for the Burr X- distribution is given by; Qu where;, log log u u Uniform. (8) That means, rom samples can be generated from the Burr X- distribution using: x X log log u where; u Uniform,. Estimation of Parameters Let,,..., n () x x x denote rom samples from the Burr X- distribution with parameters, using the method of maximum likelihood estimation (MLE), the likelihood function is given by: n exp xi exp x i exp xi,,..., n;, exp exp i exp expxi exp xi i f x x x l log f x, x,..., x ;, Let denote the log-likelihood function, then: n n n n exp l nlog nlog nlog log expxi xi i i i exp x i xi n expx i log exp i expxi () Differentiating Equation () with respect to parameters, setting the resulting non-linear system of equations to zero solving them simultaneously gives the maximum likelihood estimates of parameters respectively. It is much easier to solve these equations using algorithms in statistical software like R so on when data sets are available. III APPLICATIONS TO REAL LIFE DATA In this section, the Burr X- distribution distribution are applied to three real data. Here, judgment is based on the Log-likelihood Akaike Information Criterion () values posed by these distributions. Data I: It represents the height of female athletes (measured in cm) collected at the Australian Institute of Sport. The data has previously been used by Cook Weisberg (994), Al-Aqtash et al., (4) Owoloko et al., (6). The summary of Data I is shown in Table : Table : Summary of Data on Height of Female Athletes The performance of the Burr X- distribution with respect to its baseline distribution is as shown in Table : Table : Burr X- distribution Versus distribution (with stard error in parentheses) Likelihood Burr X e.696e.9e 4 3.4e ISBN: ISSN: (Print); ISSN: (Online) WCE 7

4 Proceedings of the World Congress on Engineering 7 Vol I WCE 7, July 5-7, 7, London, U.K. DATA II: The second data is from an accelerated life test of 59 conductors. The data has previously been used by Nasiri et al., () Oguntunde et al., (6). The observations are as follow: The data summary is as shown in Table 3: Table 3: Summary of data on accelerated life test of conductors The performance of the Burr X- distribution is assessed using Data II the result is as shown in Table 4: Table 4: Performance of Burr X- distribution using Data II (with stard error in parentheses) Likeliho od Burr X Data III: This data represents the relief times (in minutes) of patients receiving an analgesic. The data has been used recently by Shanker et al., (5) to assess the flexibility of distribution Lindley distribution. The observations are as follow: The data summary is as shown in Table 5: Table 5: Summary of data on patients receiving analgesic The performance of the Burr X- distribution is assessed using Data III the result is as shown in Table 6: Table 6: Performance of Burr X- distribution using Data III (with stard error in parentheses) Likelihood Burr X Remarks: The lower the value, the better the model the higher the log-likelihood value the better the model. IV CONCLUSION The Burr X- distribution has been successfully developed, its statistical properties like the quantile function, median, survival function, hazard function, reversed hazard function odds function have been explicitly established. The shape of the distribution has been investigated to be either unimodal or decreasing (depending on the parameter values). The model has been applied to three different data its performance was compared to the baseline distribution ( distribution), it is evident from the analysis that the Burr X- distribution failed to perform better than the baseline distribution in all the applications provided based on the log-likelihood values posed by these distributions; this result contradicts that of Yousof et al., (6) where the Burr X-Lomax distribution performed better than the Lomax distribution (its baseline distribution). ACKNOWLEDGMENT The authors would like to appreciate Covenant University for the enabling environment the anonymous referees for their constructive comments. REFERENCES [] R. Al-Aqtash, C. Lee F. Famoye. Gumbel-Weibull Distribution: Properties Applications, Journal of Modern Applied Statistical Methods, 3, -5, 4. [] A. Alzaatreh, F. Famoye C. Lee. Weibull-Pareto Distribution Its Applications, Communications in Statistics-Theory Methods, 4 (9), , 3. [3] M. Bourguignon, R. B. Silva G. M. Cordeiro. The Weibull-G Family of Probability Distributions, Journal of Data Science,, 53-68, 4. [4] R. D. Cook S. Weisberg. An Introduction to Regression Graphics (First Edition), John Wiley Sons, New York, ISBN-: 47377, pp:8, 994. [5] G. M. Cordeiro M. de Castro. A New family of Generalized Distributions, Journal of Statistical computation Simulation, 8, , [6] N. Eugene, C. Lee F. Famoye. Beta-Normal distribution Its Applications, Communications in Statistics: Theory Methods, 3, 497-5, [7] A. W. Marshall I. Olkin. A new method for adding a parameter to a family of distributions with application to the exponential weibull families, Biometrika, 84 (3), 64-65, 997. [8] P. Nasiri, I. Makhdoom B. Yaghoubian. Estimation Parameters of the Weighted Distribution, Australian Journal of Basic Applied Sciences, 5(9), 7-4,. [9] P. E. Oguntunde, E. A. Owoloko O. S. Balogun. On A New Weighted Distribution: Theory Application, Asian Journal of applied Sciences, 9(), -, 6 ISBN: ISSN: (Print); ISSN: (Online) WCE 7

5 Proceedings of the World Congress on Engineering 7 Vol I WCE 7, July 5-7, 7, London, U.K. [] E. A. Owoloko, P. E. Oguntunde A. O. Adejumo. A Comparative Analysis on the Performance of the Convoluted Distribution the Distribution in terms of Flexibility, Journal of Mathematics Statistics, (), 59-64, 6 [] W. Shaw I. Buckley. The alchemy of probability distributions: Beyond Gram-Charlier expansions a skew-kurtotic-normal distribution from a rank transmutation map. Research Report. 7 [] M. H. Tahir, G. M. Cordeiro, A. Alzaatreh, M. Mansoor M. Zubair M. The Logistic-X Family of Distributions Its Applications, Communication in Statistics-Theory Methods (To Appear), 6 [3] H. M. Yousof, A. Z. Afify, G. G. Hamedani G. Aryal G. The Burr X Generator of Distributions for Lifetime Data, 6, Retrieved from: ication/ _the_burr_x_gengenera_of_distrib utions_for_lifetime_data/links/579bf668ae8aa4 7f.pdf ISBN: ISSN: (Print); ISSN: (Online) WCE 7