Performance rating of the transmuted exponential distribution: an analytical approach

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.


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 andKundu 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.
Let X denotes a random variable, the probability density function (pdf ) and the cumulative density function (cdf ) of an Exponential distribution with parameter θ can be defined using an alternative parameterization as; and respectively.
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) 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 wellknown 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
A random variable X is said to have a transmuted distribution function if its pdf and cdf are respectively given by; where; x > 0, and | | ≤ 1 is the transmuted parameter 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. If the parameter η = 1 in Eqs. (4) and (5) of Aryal and Tsokos (2011), we have the pdf and the cdf of the TE distribution as; ( and; Respectively.
where; θ is the scale parameter λ is the transmuted parameter

Special case
For λ = 0, Eq. (5) reduces to give the pdf of the Exponential distribution. Some possible plots for the pdf of the TE distribution at some selected parameter values are shown in Figs. 1, 2, 3 4, 5 and 6; 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 | | ≤ 1.

Moments of the Transmuted Exponential distribution
Let X denote a continuous random variable, the rth moment is given by; Therefore, the rth moment of the TE distribution can be derived from; This can be obtained directly from Eq. (6) of 8 when η = 1 as; This can further be expressed as; It is obvious that for r = 1; Other higher order moments can be derived at r > 1 from Eq. (9). The table of values (at selected values) for the mean of TE distribution is provided in Table 1.

Quantile function and median of the Transmuted Exponential distribution
The quantile function x q of the TE distribution can be obtained as the inverse of Eq. (6) and in particular, when η = 1 in Eq. (7) of (Aryal and Tsokos (2011)) as; The median of the TE distribution can be obtained from Eq. (11) at q = 0.5 as; The lower quartile and upper quartile can also be derived from Eq. (11) when q = 0.25 and q = 0.75 respectively.
Plot for the pdf of TE distribution at (θ = 0.5, λ = − 0.5) Table 1 Table of   Random numbers from the TE distribution can be generated using the method of inversion; where; u ∼ U(0, 1).

Reliability analysis of the Transmuted Exponential distribution
Mathematically, the survival function is given by; Therefore, the survival function for the TE distribution can be simplified to give; The hazard function is mathematically given by; Therefore, the expression for the hazard function (or failure rate) of the TE distribution is given by; Some possible plots for the failure rate of the TE distribution at some selected parameter values are shown in Figs. 7, 8, 9 and 10;

Parameter estimation and inference for the Transmuted Exponential distribution
We make use of the method of maximum likelihood estimation (MLE) to estimate the parameters of the TE distribution. Let X 1 , X 2 , …, X n be a sample of size 'n' from the TE distribution, the likelihood function is given by; Let l = log L; Therefore; Differentiating l with respect to θ and λ respectively gives; 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.
We obtain the 2 × 2 observed information matrix through; where; The solution of the inverse matrix of the observed information matrix in Eq. (20) gives the asymptotic variance and co-variance of the maximum likelihood estimators θ and ˆ . The approximate 100 (1 − α) % asymptotic confidence interval (CI) for θ and λ are given by; where; Z α/2 is the α-th percentile of the standard normal distribution.

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.
The summary of the data is provided in Table 2; The performance of the Transmuted Exponential distribution with respect to the Beta Exponential, Generalized Exponential and Exponentiated Exponential distributions using the data on fatigue fracture is given in Table 3.
The summary of the data is provided in Table 4. The performance of the Transmuted Exponential distribution with respect to the Beta Exponential distribution, Generalized Exponential distribution and the Exponentiated Exponential distribution is as shown in Table 5.

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. There were 50 or more warnings (