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
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
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