TRANSMUTED GUMBEL UNIVARIATE EXPONENTIAL DISTRIBUTION

A functional composition of the distribution function of one probability distribution with the inverse distribution function of another is called the transmutation map. The present paper is purported to show how the transmuted distribution can be obtained by using the convex combination of failure probability of two-component systems. The transmuted Gumbel univariate exponential distribution is presented by changing convex combination parameter. This new distribution is defined and studied. Some mathematical properties of this distribution including the generating function and ordinary moments are derived. The survival, hazard rate and mean residual life functions are discussed. Finally, three applications to real data are presented.


Introduction
In the present paper, we will start by examining two-component (series and parallel) systems. The failure probabilities of these systems will be found and a new distribution is obtained by applying convex combinations to these probabilities as these can be ordered within themselves. In the process of proposing this distribution, the lifetimes of the components of the system which are the random variables are considered to be both dependent on each other and non-identical. If the random variables that represent the lifetimes of two components are identical and independent, then this proposed distribution will emerge in the transmuted model, which is one of the important families in the pertinent-literature in recent years. The transmuted family has been introduced by [27] for the …rst time and the theory of transmuted distribution is clearly de…ned by [28]. This method has led to the development of new and more ‡exible distributions by many authors, proposing many di¤erent distributions and pioneering the modeling of many real data sets with these distributions. Aryal and Tsokos [4] and [5] studied the two forms of 138 M ONIREH HAM ELDARBANDI AND M EHM ET YILM AZ the transmuted distributions. These scholars provided the mathematical characterization of transmuted extreme value and transmuted Weibull distributions and their applications to analyze real data sets. Aryal [6] proposed the transmuted loglogistic distribution and discussed various properties of this distribution. Merovci [19] introduced the transmuted Lindley distribution and applied it to bladder cancer data; Merovci [20] proposed the transmuted exponentiated exponential distribution; Merovci and Elbatal [21] studied the transmuted Lindley-geometric distribution. Ashour and Eltehiwy [7] discussed the applications of Transmuted Lomax Distribution and Ashour and Eltehiwy [8] proposed the transmuted exponentiated Lomax distribution. More recently, the transmuted exponentiated modi…ed Weibull distribution has been suggested by [13] having its applications in real data. Hussian [16] obtained the transmuted exponentiated gamma distribution and discussed their various properties and applications. Elbatal et al. [11] discussed as various estimation methods for the transmuted exponentiated Fréchet distribution. Abd El Hady [1] obtained an extended Weibull distribution as the exponentiated transmuted Weibull distribution and discussed its various properties and applications. Merovci and Puka [22] introduced the transmuted Pareto distribution. Elbatal and Aryal [12] studied the transmuted additive Weibull distribution; Merovci [23] proposed the transmuted Rayleigh distribution and discussed their various properties. In the second part of this article, the new family will be introduced and the survival and hazard rate functions of the model under study will be found. The third part of this article contains some main de…nitions as Gumbel Bivariate Exponential Distribution and Gumbel Univariate Exponential Distribution. Later, the baseline distributions of the proposed distribution will be taken as exponential distribution and the proposed distribution is called the transmuted Gumbel univariate exponential (TGUE) distribution. In the subsequent subsections, the analytical shapes of the probability density, survival, cumulative hazard rate, hazard rate and mean residual life functions of the TGUE distribution are presented. Statistical properties including moment generating function and moments, maximum likelihood estimates and the information matrix, random number generation, Rényi entropy and order statistics of the TGUE distribution are discussed in other subsections of Section 3. Finally, in order to demonstrate the usefulness of the proposed distribution, three real data applications are presented in the application section.

The New Family
In recent literature, the transmuted family of lifetime distributions have attracted the attention of the researchers for modeling the lifetime data. Firstly, two-component (series and parallel) systems will be introduced. Let T 1 and T 2 be random variables that represent the lifetime of the components. Throughout this paper, the marginal distribution functions of T 1 and T 2 are represented by F T1 (:) and F T2 (:), and the joint distribution and the joint survival functions of T 1 and T 2 are indicated by F T1;T2 (:; :) and S T1;T2 (:; :) = 1 F T1 (:) F T2 (:) + F T1;T2 (:; :), respectively. The series system success requires that the two parts operate successfully at the same time. System failure occurs if either one or more components fail. Then, the random variable T min that stands for the series system lifetime is de…ned as T min = min fT 1 ; T 2 g. Hence, the probability of the failure of the series system is given by Parallel system is such a system that functions when at least one of its components works and the failure of all the components is necessary for the system's failure to occur. Accordingly, T max = max fT 1 ; T 2 g stands for the parallel system lifetime. Then, the probability of the failure of the parallel system is given by According to axiomatic properties of probability, component lifetimes T 1 and T 2 can be ordered stochastically as T min st T i st T max ; i = 1; 2. Namely, we have P (T max t) P (T i t) P (T min t). Then, the lower and the upper bounds for F Ti (t) can be written as follows: In that case, F Ti (t) can be represented as a convex combination of failure probabilities series and parallel systems. Then, we have where the combination parameter 2 [0; 1]. This latter well-de…ned statement can derive numerous univariate distribution functions with respect to combination parameter .
In the latter equation, if the distributions of random variables T 1 and T 2 are assumed to be identical, namely, F T1 (t) = F T2 (t), then the new distribution with the parameter set is given by G (t; ) = 2 F Ti (t) + (1 2 ) F T1;T2 (t; t) : If transformation = +1 2 is done, range will change from [0; 1] to [ 1; 1]. So, for j j 1, the distribution function can be written as So, if the distributions of random variables T 1 and T 2 are taken independent, namely, F T1;T2 (t; t) = (F Ti (t)) 2 in the …rst equation of (2), we can obtain the transmuted distribution constructed by the quadratic rank transmutation method of [27] which has become very popular in the recent years.

M ONIREH HAM ELDARBANDI AND M EHM ET YILM AZ
In particular, for = 0 it gives the baseline distribution F Ti (t), for = 1, it gives the distribution of the maximum of dependent two random variables with joint distribution function F T1;T2 (t; t) and for = 1, 2F Ti (t) F T1;T2 (t; t) is the distribution of the minimum of two random variables T 1 and T 2 with identically distributed.
Theorem 1. The probability density function (p.d.f.) of T is represented in terms of the conditional hazard rates of the component lifetimes T 1 and T 2 as where 1 (t) and 2 (t) denote the failure rates of the corresponding components, given that both components are alive at time t.
Proof. The p.d.f. of this distribution can be obtained with derivation of distribution function de…ned in (2) as follows and the result in (3) will be obtained from the following method.

Survival and Hazard Rate
Functions of Proposed Distribution. The survival function denoted by S (t; ) of this distribution is de…ned as follows, The hazard rate function (hrf) corresponding to (2) and (3) is given by Thus, the hrf can be written as a weighted sum of the hrf of the random variable T 1 and sum of the conditional failure rates of the corresponding components ( 1 (t) + 2 (t)).
In the next section, we will introduce a bivariate version of the exponential distribution named the Gumbel bivariate exponential distribution. On the basis of this, the Gumbel univariate exponential distribution is de…ned and examined. Then, the transmuted Gumbel univariate exponential distribution is taken as a special case for the proposed distribution and some mathematical properties are studied.

Special Case: Transmuted Gumbel Univariate Exponential (TGUE) Distribution
We will …rst introduce distributions related to setting-up a special case. Then the baseline distribution is de…ned and we study on some reliability properties such as survival, cumulative hazard rate, hazard rate and mean residual life functions. Moment generating function and moments of proposed distribution are analyzed. ML estimation of model parameters are performed and asymptotic distribution of the parameters are obtained in terms of observed Fisher Information and then asymptotic con…dence intervals are also obtained. General expressions for the Rényi entropy is presented. Furthermore, general results for the order statistics of the TGUE random variables are derived.

Gumbel Bivariate and Univariate Exponential Distribution.
3.1.1. Gumbel Bivariate Exponential Distribution. Exponential distribution plays a central role in life testing, reliability and analyses of survival or lifetime data. The Gumbel bivariate exponential (GBE) distribution introduced by [15] is the most popular model for analyzing lifetime data and its survival function is where 1 and 2 are the scale parameters representing the characteristic life and also positive, is dependency parameter and 0 1 2 . The marginal survival functions of T 1 and T 2 respectively are e 1 t1 and e 2t2 . Hence T 1 and T 2 have exponential marginals. The p.d.f. of the three-parameter GBE distribution corresponding to (4) is given by Gumbel Univariate Exponential Distribution. By letting 1 = 2 and considering the diagonal section of S T1;T2 (t 1 ; t 2 ) i.e., t 1 = t 2 = t in the survival function of GBE distribution de…ned in (4). Then the random vector (T 1 ; T 2 ) has the Gumbel univariate exponential (GUE) distribution, and the survival function of the GUE distribution can be written as follows By using the known relation between S T1;T2 (t; t) and F T1;T2 (t; t), the distribution function of the GUE random variable is given by and its p.d.f. of the GUE random variable reduces to The moment generating function of the GUE random variable is given as follows where erf c is a complementary error function and k < . Especially, the …rst four moments of the GUE random variable T are given as 3.1.3. Transmuted Gumbel Univariate Exponential Distribution. The transmuted Gumbel univariate exponential (TGUE) distribution is an extended model to analyze more complex data. T 1 and T 2 have a exponential distribution with the same shape parameter and random vector (T 1 ; T 2 ) has a Gumbel univariate exponential distribution with and parameters, then we can write 8 < : By using equation (2) and (5), the distribution function of the TGUE random variable with the parameter space = ( ; ; ) : > 0; < 2 ; 1 1 , can be obtained as Henceforth, the p.d.f. corresponding to (3) and (6) becomes where 1 (t) = Consequently, the p.d.f. of the TGUE random variable can be written as follows g (t; ) = (1 ) e t + e (2 t+ t 2 ) ( + t + + t) ; = (1 ) e t + (2 + 2 t) e (2 t+ t 2 ) : The shapes of the p.d.f. of the TGUE random variable can be analyzed as follows by examining this derivation, it is clear that when 0 1, g 0 (t; ) < 0 is obtained and we can say that the p.d.f. is decreasing. Also, in order for p.d.f. to be unimodal, it must be 1 0.
Thus, we …nd the cumulative hazard function of the TGUE random variable and this function describes how the risk of a particular outcome changes with time. We know H (0; ) = 0; lim is increasing for all t 0.
The other characteristic of a random variable is the hrf. By using (7) and (8), this function is given as follows The hrf of the TGUE random variable has the following properties: h (0; ) = (1 + ) ; (1 ) e t + (2 + 2 t) e (2 t+ t 2 ) (1 ) e t + e (2 t+ t 2 ) = ; (2 + 2 t) = 1: The hazard rate function will be examined in the extreme values of the parameters: (1) If = 0, the hrf is the same as the exponential distribution; h (t; ) = (2) If = 1, the hrf is the same as the linear hazard rate function; h (t; ) = 2 ( + t) (3) If = 0, the hrf is the same as the transmuted exponential distribution; Let's investigate the monotonicity of hrf, (1 ) e t + e (2 t+ t 2 ) 2 : It is clear from above derivation, when 1 0, the hazard rate function is increasing, that is, h 0 (t; ) 0. When 0 1, the hazard rate function is decreasing (h 0 (t; ) 0). Some possible shapes of hrf for selected parameter value are shown in the following …gures. Figure 3.2 shows the hrf de…ned in (10) with di¤erent choices of parameters. This distribution has an increasing hrf for 1 0. If 0 1, the hrf is decreasing.

Mean Residual Life Function of the TGUE Random Variable.
In this section, we will …nd the mean residual life (mrl) function of the TGUE random variable which is another important characteristic of a random variable.
The mrl function of the TGUE random variable has the following properties: (1) If = 0, the mrl function is the same as the exponential distribution; and some possible shapes of (11) for selected parameter values is showed in the following …gures.
3.1.6. 3.5. Moment Generating Function and moments of the TGUE Random Variable. In this section, we derive the moment generating function and …rst four moments for the TGUE distribution. Let T have the TGUE distribution, then the moment generating function of T is given by The expressions for the expected value and variance are 3.1.7. Estimation by Maximum Likelihood and the Information Matrix of the TGUE Distribution. Let (t 1 ; t 2 ; ; t n ) be sample values from this distribution with parameters ; and . The likelihood function for = f ; ; g is given by (1 ) e ti + (2 + 2 t i ) e (2 ti+ t 2 i ) : Throughout this subsection, the log-likelihood function is denoted by l = log L ( ; t 1 ; t 2 ; ; t n ) for brevity. We di¤erentiate l with respect to ; and as follows The maximum likelihood estimators as^ ,^ and^ are obtained by equating these three equations (12), (13) and (14) to zero and solving the equations simultaneously. For these three parameters, we will get the second order derivatives of logarithms of the likelihood function for obtaining the elements of the Fisher-Information Matrix.
Thus, Fisher information matrix, I n ( ) of sample size n for is as follows:  I  I  I  I  I  I  I  I  I   1 A Inverse of the Fisher-information matrix of single observation, i.e., I 1 1 ( ) indicates asymptotic variance-covariance matrix of maximum likelihood estimates of . Hence, the distribution of maximum likelihood estimator for is asymptotically normal with mean and variance-covariance matrix I 1 1 ( ). Namely, 2 4^ ^ ^ By solving this inverse dispersion matrix these solutions will yield asymptotic variance and covariance of these ML estimators for these parameters.
We can approximate 100 (1 ) % con…dence intervals for , and by using (15)  where z 1 2 is the upper 100 the quantile of the standard normal distribution, and I 1 1 denotes respective diagonal elements of I 1 1 :

Random Number Generation from the TGUE Distribution.
Remember the distribution function de…ned in section 2, where 0 1. Again, emphasize that G (t) represents a two-component mixture distribution, where the distribution functions of the T min and T max are the components of this mixture, respectively. To generate a random number from G(t), we apply the reference Gentle [14] pp.125. Accordingly, a random number V is generated from uniform distribution on (0,1) to decide which of the components are chosen. As a result, when V , the random number will be generated from F Tmin (t) by equating as F Tmin (t) = V . Otherwise, namely V > , the random number will be generated from the distribution of T max by equating F Tmax (t) = V .
First of all, we will consider how to produce component lifetimes. By citing the method given in Gentle [14] pp.109, these component lifetimes will be generated with the help of the conditional distribution function. Namely, F T1;T2 (t 1 ; t 2 ) can be expressed as the product of the cdf of T 1 and the conditional cdf of T 2 with given In the …rst step, a random number U 1 is generated from the uniform distribution on the interval (0; 1). Then we generate the lifetime of the …rst component t 1 = F 1 T1 (U 1 ). In the second step, again we generate a uniformly distributed random variable U 2 (independent of U 1 ) on (0; 1). Therefore, the lifetime of the second component can be generated by equating t 2 = F 1 T2jT1=t1 (U 2 ). Hence, the random number from the TGUE is generated as for V , t = 150 M ONIREH HAM ELDARBANDI AND M EHM ET YILM AZ minâft 1 ; t 2 g and for V > , t = maxâft 1 ; t 2 g. Then, according to the abovementioned steps, t 1 = 1 ln (1 U 1 ) and t 2 = Here W 1 (:) denotes the lower part of Lambert W-function whose domain is e 1 ; 0 and range ( 1; 1]. A more detailed inference about generating second component lifetime is given in the appendix. 3.1.9. Rényi Entropy of the TGUE Distribution. The entropy of a random variable is a measure of variation of the uncertainty, see [25]. Then the Rényi entropy function of the random variable T with p.d.f. (7) is de…ned by where > 0, 6 = 1. We have the following series representation of (g (t; )) by applying the generalized Binomial theorem to obtain Rényi entropy for proposed distribution. Accordingly, (g (t; )) = (1 ) e t + (2 + 2 t) e (2 t+ t 2 ) : (g (t; )) can be written as an in…nite series representation as follows.
In the latter equation, the statement e ( +j) t j t 2 is rearranged as; and if the Binomial theorem is applied in (2 + 2 t) j , we can write Then, the Rényi entropy can be written as follows (1 ) e t + e (2 t+ t 2 ) n j ; therefore, the p.d.f. of the …rst order statistics T (1) is given by i n 1 ; and the p.d.f. of the last order statistics T (n) is given Note that = 0 yields the order statistics of the exponential distribution with parameter and when = 1 yields the order statistics of the TGUE distribution with parameter ( ; ).

Numerical Examples
In this section, we provide three data analyses in order to assess the goodness-of-…t of the TGUE distribution. The following tables show goodness-of-…t measures for the di¤erent distributions.  27.1, 20.2, 16.8, 5.3, 9.7, 27.5, 2.5, 27.0, 1.9, 2.8. Firstly, these data were analyzed by [10]. Later on, Beta-Pareto (BP) distribution was applied to these data by [2]. Merovcia and Pukab [22] made a comparison between Pareto (P) and Transmuted Pareto (TP) distribution. They showed that better model is the transmuted Pareto distribution. Bourguignon et al. [9] proposed Kumaraswamy Pareto (Kw-P) distribution. Tahir [30] have proposed Weibull-Pareto (WP) distribution and made a comparison with Beta Exponentiated Pareto (BEP) distribution. Nasiru and Luguterah [24] have proposed a di¤erent type of Weibull-Pareto (NWP) distribution. Exponential Modi…ed Discrete Lindley (EMDL) distribution was applied to these data by [31]. We …t data to TGUE distribution and get parameter estimates as^ = 0:0672;^ = 0:2972;^ = 0:1976• { . According to the model selection criteria (AIC) tabulated in Table 5.1, TGUE takes the …rst place amongst 9 proposed models.   [3] and was also used by [29]. They …t this data to Lindley (L) and generalized Lindley (GL) distributions. We …t data to TGUE distribution and get parameter estimates aŝ = 0:185;^ = 0:472;^ = 0:222. According to the model selection criteria tabulated in Table 5.3, it is said that TUGE takes …rst place in amongst 3 proposed models. In the above three tables, it is clear that the values of the Akaike information criterion (AIC) and Bayesian information criterion (BIC) are smaller for the TGUE distribution compared to those values of the other models; the new distribution is a very competitive model to these data.

Conclusion
In this article, we propose a new model of transmuted distribution so-called the transmuted Gumbel univariate exponential distribution. The subject distribution is generated by using the convex combination of failure probabilities of two-component series and systems and taking the Gumbel univariate exponential distribution as the base distribution. Some mathematical and statistical properties including explicit expressions for the probability density, survival, cumulative hazard rate, hazard rate and mean residual life functions, also, moment generating function and moments are addressed. The estimation of parameters is approached by the maximum likelihood method. According to K-S values in Numerical Examples Section, the applications of the transmuted Gumbel univariate exponential distribution to real data show that the new distribution can be used to provide better …ts than the other distributions. We hope that this new distribution may attract wider applications in the lifetime literature. Taking bivariate distributions will guide to derivation of many new univariate distributions.

Appendix
Conditional cdf of T 2 with given T 1 = t 1 is given by F T2jT1 (t 2 ) = @ @t1 F T1;T2 (t 1 ; t 2 ) f T1 (t 1 ) = @ @t1 1 e t1 e t2 + e (t1+t2) t1t2 e t1 = e t1 ( + t 2 ) e (t1+t2) t1t2 e t1 = 1 1 + t 2 e ( + t1)t2 : Hence, by equating F T2jT1 (t 2 ) = U 2 where U 2 is uniformly distributed random variable on the interval (0; 1) we have a non linear equation to get solution for t 2 as follows, To solve the above equation for t 2 , we use Lambert W-function which is de…ned as the solution of the equation W (z) e W (z) = z, where z is the complex number. If z is any real number, then this equation has a solution on e 1 ; +1) . In equation (17), if the expression 1 + t 2 is taken as z, we can write ze 2 + t1 z e 2 + t1 = 1 U 2 : Multiplying both sides of equation above by 2 + t 1 , above expression can be simpli…ed as follows, where = (1 U 2 ) 2 + t 1 e 2 + t1 . Hence, the solution for W (z) is So, t 2 is found as follows To show the uniqueness of the solution for t 2 we take into account the well known inequality e (z+1) z and replacing z with ( + t 1 ) , then