- Research
- Open Access
Performance rating of the transmuted exponential distribution: an analytical approach
- Enahoro A. Owoloko^{1},
- Pelumi E. Oguntunde^{1}Email author and
- Adebowale O. Adejumo^{2}
- Received: 10 September 2015
- Accepted: 4 December 2015
- Published: 24 December 2015
Abstract
In this article, the so called Transmuted Exponential (TE) distribution was applied to two real life datasets to assess its potential flexibility over some other generalized models. Various statistical properties of the TE distribution were also identified while the method of maximum likelihood estimation was used to estimate the model parameters.
Keywords
- Estimation
- Flexibility
- Maximum likelihood estimation
- Properties
- Transmuted Exponential
Background
Attempts to generalize the Exponential distribution have led to the developement of Beta Exponential distribution (Nadarajah and Kotz 2006), Kumaraswamy Exponential distribution (Cordeiro and de Castro 2011), Generalized Exponential distribution (Gupta and Kundu 1999, 2007) and Exponentiated Exponential distribution (Gupta 2001). These distributions have been found to be more flexibly than the Exponential distribution when applied to real life data sets.
where; θ is the scale parameter
Several generalized families of distributions have been proposed in the literature, for instance, the β-G; (Eugene et al. 2002), Kumaraswamy-G; (Cordeiro and de Castro 2011), Transmuted family of distributions; (Shaw and Buckley 2007), Gamma-G (type 1); (Zografos and Balakrishnan 2009), McDonald-G; (Alexander et al. 2012), Gamma-G (type 2); (Ristic et al. 2012), Gamma-G (type 3); (Torabi and Montazari 2012), Log-gamma-G; Amini et al. (2012), Exponentiated T-X; Alzaghal et al. (2013), Exponentiated-G (EG); (Cordeiro et al. 2013), Logistic-G; Torabi and Montazari (2014), Gamma-X; (Alzaatreh et al. 2013), Logistic-X; (Tahir et al. 2015), Weibull-X; (Alzaatreh et al. 2013), Weibull-G; (Bourguignon et al. 2014) and Beta Marshall-Olkin family of distributions; (Alizadeh et al. 2015) and many others are available in the literature.
Of interest to us in this article is the Transmuted family of distribution which was obtained using the quadratic rank transmutation map. The transmuted family of distributions has been adopted by several notable authors to generalize known theoretical models, the Transmuted Weibull distribution; Aryal and Tsokos (2011), Transmuted Rayleigh distribution; (Merovci 2013), Transmuted Exponentiated Modified Weibull distribution; (Ashour and Eltehiwy 2013a), Transmuted Modified Weibull distribution; Khan and King (2013), Transmuted Lomax distribution; (Ashour and Eltehiwy 2013b), Transmuted Exponentiated Gamma distribution; Hussian (2014), Transmuted Inverse Rayleigh distribution; Ahmad et al. (2014), Transmuted Pareto distribution; (Merovci and Puka 2014), Transmuted Inverse Weibull distribution; (Khan et al. 2014), Transmuted Modified Inverse Weibull Distribution; (Elbatal 2013), Transmuted Additive Weibull distribution; (Elbatal and Aryal 2013), Transmuted Complementary Weibull Geometric Distribution; (Afify et al. 2014), Transmuted Inverse Exponential distribution; (Oguntunde and Adejumo 2015), Transmuted Size-Biased Exponential distribution; Ahmad et al. (2015) and Transmuted Gompertz distribution; (Abdul-Moniem and Seham 2015); are some known examples in the literature.
The aim of this article is to obtain the Transmuted Exponential (TE) distribution as a special case of Transmuted Weibull distribution following the content of Aryal and Tsokos (2011) and to assess its flexibility over some other generalized models using real life data sets.
The rest of this article is organized as follows; in "The Transmuted Exponential (TE) distribution: existing and more results", the TE distribution, its properties and various statistical properties are discussed, real life applications with respect to some other well-known generalized models shall be discussed in "Application", followed by concluding remark. The R-code for the analysis is provided as “Appendix”.
The Transmuted Exponential (TE) distribution: existing and more results
G(x) is the cdf of the baseline distribution.
f(x) and g(x) are the associated pdf of F(x) and G(x), respectively.
When λ = 0; Eqs. (3) and (4) reduces to the baseline distribution.
Respectively.
For x > 0, θ > 0, \(\left| {\lambda \le 1} \right|\)
where;
θ is the scale parameter
λ is the transmuted parameter
Special case
Depending on the parameter values, it can be observed from the figures above that the shape of the TE distribution could be decreasing, or inverted bathtub (unimodal). It should also be noted that \(\left| \lambda \right| \le 1\).
Moments of the Transmuted Exponential distribution
Table of means for the Transmuted Exponential distribution
λ = −0.1 | λ = −0.4 | λ = −0.7 | λ = −1.0 | λ = 0 | λ = 0.1 | λ = 0.4 | λ = 0.7 | λ = 1.0 | |
---|---|---|---|---|---|---|---|---|---|
θ = 1 | 1.05 | 1.20 | 1.35 | 1.50 | 1.00 | 0.95 | 0.80 | 0.65 | 0.50 |
θ = 2 | 2.10 | 2.40 | 2.70 | 3.00 | 2.00 | 1.90 | 1.60 | 1.30 | 1.00 |
θ = 3 | 3.15 | 3.60 | 4.05 | 4.50 | 3.00 | 2.85 | 2.40 | 1.95 | 1.50 |
θ = 4 | 4.20 | 4.80 | 5.40 | 6.00 | 4.00 | 3.80 | 3.20 | 2.60 | 2.00 |
θ = 5 | 5.25 | 6.00 | 6.75 | 7.50 | 5.00 | 4.75 | 4.00 | 3.25 | 2.50 |
θ = 6 | 6.30 | 7.20 | 8.10 | 9.00 | 6.00 | 5.70 | 4.80 | 3.90 | 3.00 |
θ = 7 | 7.35 | 8.40 | 9.45 | 10.50 | 7.00 | 6.65 | 5.60 | 4.55 | 3.50 |
θ = 8 | 8.40 | 9.60 | 10.80 | 12.00 | 8.00 | 7.60 | 6.40 | 5.20 | 4.00 |
θ = 9 | 9.45 | 10.80 | 12.15 | 13.50 | 9.00 | 8.55 | 7.20 | 5.85 | 4.50 |
θ = 10 | 10.50 | 12.00 | 13.50 | 15.00 | 10.00 | 9.50 | 8.00 | 6.50 | 5.00 |
Quantile function and median of the Transmuted Exponential distribution
The lower quartile and upper quartile can also be derived from Eq. (11) when q = 0.25 and q = 0.75 respectively.
Reliability analysis of the Transmuted Exponential distribution
Parameter estimation and inference for the Transmuted Exponential distribution
Equating Eqs. (18) and (19) to zero and solving the resulting nonlinear system of equations gives the maximum likelihood estimates of parameters θ and λ.
Application
The models to be compared in this section include the TE distribution, Beta Exponential distribution, Generalized Exponential Distribution and the Exponentiated Exponential distribution. The analyses were performed with the aid of R software.
Data Set I. The first data represents the life of fatigue fracture of Kevlar 373/epoxy subjected to constant pressure at 90 % stress level until all had failed. The data was extracted from (Abdul-Moniem and Seham 2015) and it has previously been used by Barlow et al. (1984). The data is as follows;
0.0251, 0.0886, 0.0891, 0.2501, 0.3113, 0.3451, 0.4763, 0.5650, 0.5671, 0.6566, 0.6748, 0.6751, 0.6753, 0.7696, 0.8375, 0.8391, 0.8425, 0.8645, 0.8851, 0.9113, 0.9120, 0.9836, 1.0483, 1.0596, 1.0773, 1.1733, 1.2570, 1.2766, 1.2985, 1.3211, 1.3503, 1.3551, 1.4595, 1.4880, 1.5728, 1.5733, 1.7083, 1.7263, 1.7460, 1.7630, 1.7746, 1.8275, 1.8375, 1.8503, 1.8808, 1.8878, 1.8881, 1.9316, 1.9558, 2.0048, 2.0408, 2.0903, 2.1093, 2.1330, 2.2100, 2.2460, 2.2878, 2.3203, 2.3470, 2.3513, 2.4951, 2.5260, 2.9911, 3.0256, 3.2678, 3.4045, 3.4846, 3.7433, 3.7455, 3.9143, 4.8073, 5.4005, 5.4435, 5.5295, 6.5541, 9.0960.
Summary of data on fatigue fracture of Kevlar 373/epoxy at 90 % stress level (to four decimal places)
Min. | Q_{1} | Q_{2} | Q_{3} | Mean | Max. | Variance | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|---|
0.0251 | 0.9048 | 1.7360 | 2.2960 | 1.9590 | 9.0960 | 2.4774 | 1.9406 | 8.1608 |
Performance rating of selected models
Distributions | Estimates | Log-likelihood | AIC |
---|---|---|---|
Transmuted Exponential (θ, λ) | θ = 1.3763, λ = −0.8487 | −121.5166 | 247.0331 |
Beta Exponential (a, b, θ) | a = 1.6797, b = 1.5085, θ = 0.4849 | −122.2275 | 250.4551 |
Generalized Exponential (a, θ) | a = 1.70949, θ = 0.70279 | −122.2436 | 248.4872 |
Exponentiated Exponential (a, θ) | a = 39.969318, θ = 0.012770 | −127.1143 | 258.2287 |
Data Set II. The second data set represents the monthly actual taxes revenue (in 1000 million Egyptian pounds) in Egypt between January 2006 and November 2010. The data was extracted from Nassar and Nada (2011). The data is as follows;
5.9, 20.4, 14.9, 16.2, 17.2, 7.8, 6.1, 9.2, 10.2, 9.6, 13.3, 8.5, 21.6, 18.5, 5.1, 6.7, 17, 8.6, 9.7, 39.2, 35.7, 15.7, 9.7, 10, 4.1, 36, 8.5, 8, 9.2, 26.2, 21.9, 16.7, 21.3, 35.4, 14.3, 8.5, 10.6, 19.1, 20.5, 7.1, 7.7, 18.1, 16.5, 11.9, 7, 8.6, 12.5, 10.3, 11.2, 6.1, 8.4, 11, 11.6, 11.9, 5.2, 6.8, 8.9, 7.1, 10.8.
Summary of data on tax revenue (to two decimal places)
Min. | Q_{1} | Q_{2} | Q_{3} | Mean | Max. | Variance | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|---|
4.10 | 8.45 | 10.60 | 16.85 | 13.49 | 39.20 | 64.83 | 1.57 | 5.26 |
Performance rating of selected models
Distributions | Estimates | Log-likelihood | AIC |
---|---|---|---|
Transmuted Exponential (θ, λ) | θ = 3.862 × 10^{5}, λ = 9.389 × 10^{−4} | −83.44494 | 170.8899 |
Beta Exponential (a, b, θ) | a = 63.52239, b = 0.16957, θ = 0.76882 | −187.9398 | 381.8795 |
Generalized Exponential (a, θ) | a = 5.53040, θ = 0.17867 | −191.2235 | 386.4471 |
Exponentiated Exponential (a, θ) | a = 11.755728, θ = 0.006307 | −212.5068 | 429.0136 |
Discussion
The model corresponding to the lowest Akaike Information Criteria (AIC) or the highest Log-likelihood value is regarded as the ‘best’ model. In this case, the TE distribution has the lowest AIC value with 247.0331 and 170.8899 respectively. Also, it has the highest value of Log-likelihood of −121.5166 and −83.44494 respectively. Hence, it can be regarded as a better model for the data used.
Conclusion
This article studies the performance of the TE distribution with respect to some other generalized models. The shape of the TE distribution could be decreasing or unimodal (depending on the value of the parameters). The TE distribution appeared to be better than the Beta Exponential distribution, Generalized Exponential distribution and the Exponentiated Exponential distribution in terms of flexibility when applied two real life data. The criteria used are the Log-likelihood value and the AIC.
Declarations
Authors’ contributions
OPE is a research student in the Department of Mathematics, Covenant University under the supervision of Dr. AOA and Dr. EAO. He developed the idea that led to this article. The supervisors guided, read through and all agreed with the results and findings. All authors read and approved the final manuscript.
Acknowledgements
The authors appreciate the anonymous referees for their useful and timely comments towards improving the quality of this paper. The financial support from Covenant University is also deeply appreciated.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Abdul-Moniem IB, Seham M (2015) Transmuted Gompertz distribution. Comput Appl Math J 1(3):88–96Google Scholar
- Afify AZ, Nofal ZM, Butt NS (2014) Transmuted complementary Weibull geometric distribution. Pak J Stat Operation Res 10(4):435–454View ArticleGoogle Scholar
- Ahmad A, Ahmad SP, Ahmed A (2014) Transmuted Inverse Rayleigh distribution: a generalization of the Inverse Rayleigh distribution. Math Theory Model 4(7):90–98Google Scholar
- Ahmad K, Ahmad SP, Ahmed A (2015) On new method of estimation of parameters of transmuted size-biased exponential distribution and its structural properties. Int J Innov Res Stud 4(3):1–12Google Scholar
- Alexander C, Cordeiro GM, Ortega EMM, Sarabia JM (2012) Generalized beta-generated distributions. Comput Stat Data Anal 56:1880–1897View ArticleGoogle Scholar
- Alizadeh M, Cordeiro GM, de Brito E, Demetrio CGB (2015) The beta Marshall-Olkin family of distributions. J Stat Distrib Appl 2(4):1–18View ArticleGoogle Scholar
- Alzaatreh A, Famoye F, Lee C (2013) Weibull–Pareto distribution and its applications. Commun Stat-Theory Methods 42(9):1673–1691View ArticleGoogle Scholar
- Alzaghal A, Lee C, Famoye F (2013) Exponentiated T-X family of distributions with some applications. Int J Probab Stat 2:31–49View ArticleGoogle Scholar
- Amini M, MirMostafaee SMTK, Ahmadi J (2012) Log-gamma-generated families of distributions. Statistics. doi:10.1008/02331888.2012.748775 Google Scholar
- Aryal GR, Tsokos CP (2011) Transmuted Weibull distribution: a genaralization of the Weibull probability distribution. Eur J Pure Appl Math 4:89–102Google Scholar
- Ashour SK, Eltehiwy MA (2013a) Transmuted exponentiated modified Weibull distribution. Int J Basic Appl Sci 2(3):258–269Google Scholar
- Ashour SK, Eltehiwy MA (2013b) Transmuted Lomax distribution. Am J Appl Math Stat 1(6):121–127View ArticleGoogle Scholar
- Barlow RE, Toland RH and Freeman T (1984) A Bayesian analysis of stress rupture life of Kevlar 49/epoxy spherical pressure vessels, In: Procedings. conference on applications of statistics, Marcel Dekker, New YorkGoogle Scholar
- Bourguignon M, Silva RB, Cordeiro GM (2014) The Weibull-G family of probability distributions. J Data Sci 12:53–68Google Scholar
- Cordeiro GM, de Castro M (2011) A New family of generalized distributions. J Stat Comput Simul 81:883–898View ArticleGoogle Scholar
- Cordeiro GM, Ortega EMM, da Cunha DCC (2013) The exponentiated generalized class of distribution. J Data Sci 11:1–27Google Scholar
- Elbatal I (2013) Transmuted modified inverse Weibull Distribution: a generalization of the modified inverse Weibull probability distribution. Int J Math Arch 4(8):117–129Google Scholar
- Elbatal I, Aryal G (2013) On the transmuted additive Weibull distribution. Aust J Stat 42(2):117–132Google Scholar
- Eugene N, Lee C, Famoye F (2002) Beta-normal distribution and its applications. Commun Stat: Theory Methods 31:497–512View ArticleGoogle Scholar
- Gupta RD (2001) Exponentiated exponential family; an alternative to gamma and Weibull. Biometrical J 33:117–130View ArticleGoogle Scholar
- Gupta RD, Kundu D (1999) Generalized exponential distributions. Aust N Z J Stat 41:173–188View ArticleGoogle Scholar
- Gupta RD, Kundu D (2007) Generalized exponential distribution: existing methods and recent developments. J Stat Plann Inference 137:3537–3547View ArticleGoogle Scholar
- Hussian MA (2014) Transmuted exponentiated gamma distribution; a generalization of the exponentiated gamma probability distribution. Appl Math Sci 8(27):1297–1310Google Scholar
- Khan MS, King R (2013) Transmuted modified Weibull distribution: a generalization of the modified Weibull probbility distribution. Eur J Pure Appl Math 6:66–88Google Scholar
- Khan MS, King R, Hudson I (2014) Characteristics of the transmuted inverse Weibull distribution. ANZIAM J. 55(EMAC2013):C197–C217Google Scholar
- Merovci F (2013) The transmuted rayleigh distribution. Aust J Stat 22(1):21–30Google Scholar
- Merovci F, Puka I (2014) Transmuted pareto distribution. ProbStat Forum 07:1–11Google Scholar
- Nadarajah S, Kotz S (2006) The Beta exponential distribution. Reliability Eng Syst Saf 91(6):689–697View ArticleGoogle Scholar
- Nassar MM, Nada NK (2011) The beta generalized pareto distribution. J Stat: Adv Theory Appl 6(1/2):1–17Google Scholar
- Oguntunde PE, Adejumo AO (2015) The transmuted inverse exponential distribution. Int J Adv Stat Probab 3(1):1–7View ArticleGoogle Scholar
- Risti´c, Miroslav M, Balakrishnan N (2012) The gamma-exponentiated exponential distribution. J Stat Comput Simulation 82:1191–1206View ArticleGoogle Scholar
- Shaw WT, Buckley IR (2007) The alchemy of probability distributions: beyond Gram-Charlier expansions and a skew-kurtotic-normal distribution from a rank transmutation map. Research reportGoogle Scholar
- Tahir MH, Cordeiro GM, Alzaatreh A, Mansoor M, Zubair M (2015) The logistic-X family of distributions and its applications. Commun Stat-Theory Methods (forthcoming)Google Scholar
- Torabi H, Montazari NH (2012) The gamma-uniform distribution and its application. Kybernetika 48:16–30Google Scholar
- Torabi H, Montazari NH (2014) The logistic-uniform distribution and its application. Commun Stat-Simul Comput 43:2551–2569View ArticleGoogle Scholar
- Zografos K, Balakrishnan N (2009) On families of beta- and generalized gamma-generated distributions and associated inference. Stat Methodol 6:344–362View ArticleGoogle Scholar