New Generalizations of Exponential Distribution with Applications

The gamma family distributions was discussed by Karl Pearson in 1895 as pointed out in Balakrishnan and Basu [1]. However, after a period of 35 years the Exponential Distribution is a special case of the gamma distribution to appear on its own. It is also related to Poisson process as it has been observed that the time between two successive Poisson events follows the Exponential Distribution. While discussing the sampling of standard deviation (SD), the Exponential Distribution was referred to by Kondo [2] as Pearson’s TypeX distribution. Steffensen [3], Teissier [4], and Weibull [5] proposed the applications of Exponential Distribution in actuarial, biological, and engineering problems, respectively. An extension of Exponential Distribution was proposed by Weibull (1951). The Exponential Distribution is a special case wherein the shape parameter equals one. The Weibull distribution has many applications in survival analysis and reliability engineering; for reference see Lai et al. (2006). Some other applications in industrial quality control are discussed in Berrettoni (1964). 2. Some Basic Definitions and K-Generalized Exponential Distributions


Introduction
The gamma family distributions was discussed by Karl Pearson in 1895 as pointed out in Balakrishnan and Basu [1].However, after a period of 35 years the Exponential Distribution is a special case of the gamma distribution to appear on its own.It is also related to Poisson process as it has been observed that the time between two successive Poisson events follows the Exponential Distribution.While discussing the sampling of standard deviation (SD), the Exponential Distribution was referred to by Kondo [2] as Pearson's Type  distribution.Steffensen [3], Teissier [4], and Weibull [5] proposed the applications of Exponential Distribution in actuarial, biological, and engineering problems, respectively.
An extension of Exponential Distribution was proposed by Weibull (1951).The Exponential Distribution is a special case wherein the shape parameter equals one.The Weibull distribution has many applications in survival analysis and reliability engineering; for reference see Lai et al. (2006).Some other applications in industrial quality control are discussed in Berrettoni (1964).

Some Basic Definitions and 𝐾-Generalized Exponential Distributions
We begin with some definitions which provide a base for the definition of -Generalized Exponential Distributions.
The Euler gamma function Γ() is defined by the integral A random variable  is said to have gamma distribution with parameter  > 0, if its p.d.f. is given by  () = 1 Γ ()  −1  − , 0 <  < ∞  () = 0, elsewhere. ( Replacing  by /, we get the following form of gamma distribution with parameters ,  with  > 0 and  > 0: −1  −/ , 0 <  < ∞,  > 0,  > 0. ( The gamma distribution with parameters ,  often arises in practices, as the distribution of time one has to wait until a fixed number of events have occurred.
The beta function of two variables  and  is defined by In the literature (for reference see [6][7][8]), it is known that In ( 5), if we take then 0 ≤  ≤ 1 implies 0 ≤  < ∞, and we get A continuous random variable  is said to have beta distribution with parameters  and , if its p.d.f. is given by This distribution is known as beta distribution of 1st kind (for reference see [7]).
The beta distribution has an application to model a random phenomenon whose set of possible values is a finite interval [, ], which by letting  denote the origin and taking ( − ) as a unit measurement can be transformed into the interval [0, 1].
For  > 0 and  ∈ L, the -gamma function is defined by the integral For () > 0, () > 0, the -beta function of two variables  and  is defined by It is implicit in the literature (for reference see [7]) that For the sake of completeness, we present a very simple proof of the relation (13).
We have Put then and we get Now Put or so that we get Since the integrals involved are convergent, we have Put so that we get Using ( 17) and ( 22), we get which gives This completes the proof of the relation (13).
A random variable  of continuous type is said to have Weibull distribution if its probability density function is given by Weibull distribution is widely used in engineering practice due to its versatility.It was originally proposed for the interpretation of fatigue data but now it is also used for many other problems in engineering.In particular, in the field of life phenomenon, it is used as the distribution of lifetime of some object, particularly when the "weakest link" model is appropriate for the model; that is, consider an object consisting of many parts and suppose that the object experiences death (failure) when any of its parts fail; it has been shown [6] (both oretically and empirically) under these conditions that Weibull distribution provides a close approximation to the distribution of the lifetime of the item.
The Gamma and Weibull distributions are commonly used for analyzing any lifetime data or skewed data.Both distributions have nice physical interpretation and several desirable properties.Unfortunately both distributions have drawbacks, one major disadvantage of the gamma distribution is that the distribution function or the survival function can not be computed easily if the shape parameter is not an integer.By using mathematical tables or computer software one obtains the distribution function, the survival function, or hazard function.This makes the gamma distribution unpopular as compared to Weibull distribution whose distribution function, hazard function, or survival function is easy to compute.It is well known that even though the Weibull distribution has convenient representation of distribution function, the distribution of the sum of independent and identically distributed (i.i.d) Weibull random variables is not simple to obtain.Therefore, the distribution of the mean of random sample from Weibull distribution is not easy to compute whereas the distribution of sum of independent and identically distributed (i.i.d) gamma random variables is well known.For more details see Mudholkar, Srivastava and Freimer (1995), Mudholkar and Srivastava [17], Gupta and Kundu [18], and Gupta et al. [19].
Recently R. D. Gupta and D. Kundu have introduced three-parameter Exponential Distribution (location, scale, and shape) and studied the theoretical properties of this family and compared them with respective good studies properties of Gamma and Weibull distributions.The increasing and decreasing hazard rate of the Generalized Exponential Distribution (GED) depends on the shape parameter.Generalized Exponential Distribution (GED) has several properties that are quite similar to gamma distribution but it has distribution function similar to that of the Weibull distribution which can be computed easily.Since the Generalized Exponential family has the likelihood ratio ordering on the shape parameter, one can construct a uniformly most powerful test for testing one sided hypothesis on the shape parameter when the scale and location parameters are known.

Generalized Exponential Distribution (GED)
A continuous random variable  whose p.d.f. is given by is said to have an Exponential Distribution.

Extensions of Generalized Exponential Distribution (GED)
The main aim of this paper is to present interesting extensions of Generalized Exponential Distribution (GED) in various ways and to study their moment generating functions (MGFs).We shall first define Generalized Exponential Distribution (GED) in terms of a new parameter  > 0 and call it -Generalized Exponential Distribution (-GED).In fact, we prove the following result, which included Generalized Exponential Distribution as a special case.
Theorem 2. Let  be a random variable of continuous type and let  > 0,  > 0, and  > 0 be the parameters; then the function is the p.d.f. of random variable  of continuous type.
Remark 3. If we take  = 1, it reduces to Generalized Exponential Distribution.

The Moment Generating Function (MGF) of Theorem 2
In this section, we derive MGF of the random variable  having k-Generalized Exponential Distribution in terms of new parameter  > 0; we have Therefore, Our next theorem is also a generalization of Exponential Distribution in terms of new variable , which includes Weibull distribution as a special case.
Hence (, , , ) is a p.d.f. of random variable  of continuous type.

The Moment Generating Function (MGF) of Theorem 4
In this section, we derive MGF of the random variable  having k-Generalized Exponential Distribution in terms of a new parameter  > 0; we have Put  −  = ; then Therefore, We also present the following three-parameter extension of Generalized Exponential Distribution (GED).In fact, we prove the following.Theorem 7. Let  be a continuous random variable; then the function is the p.d.f. of random variable  of continuous type.

The Moment Generating Function (MGF) of Theorem 7
where Remark 10.For  = 1, we have (67) Remark 12. Letting  → 0 in (57) and noting that we get which is the p.d.f. of the Exponential Distribution.
Finally we present the following more general interesting result which among other things includes Weibull distribution as a limiting case.
This shows that (, , , ) is the p.d.f. of random variable  of continuous type.This proves Theorem 13.
Remark 14. Weibull Distribution is the limiting case of Theorem 13.To see this, we let  → 0 in Theorem 13 and note that lim so that is the p.d.f. of random variable  of continuous type.
Replacing  by ( − ])/, it follows that Equivalently is a p.d.f. of , which is clearly density of Weibull Distribution.

Applications
The applications of Exponential Distribution have been widespread, which include models to determine bout criteria for analysis of animal behaviour [20]; design rainfall estimation in the Coast of Chiapas [21]; analysis of Los Angeles rainfall data [22]; software reliability growth models for vital quality metrics [23]; models for episode peak and duration for ecohydroclimatic applications [24]; estimating mean life of power system equipment with limited end-oflife failure data [25]; and cure rate modeling (Kannan et al. (2010).In related work, the closeness of the exponentiated Exponential Distribution with the Weibull, gamma, and lognormal distributions is studied in Gupta and Kundu (2003a), (2003b), ( 2004), ( 2006)) and [26].Some generalizations of the exponentiated Exponential Distribution are discussed in Nadarajah and Kotz [27].The Exponential Distribution is often used to model the reliability of electronic systems, which do not typically experience wear-out type failures.The distribution is called "memoryless," meaning that the calculated reliability for, say, a 10-hour mission is the same for a subsequent 10-hour mission, given that the system is working properly at the start of each mission.Given a hazard (failure) rate, , or mean time between failure (MTBF = 1/), the reliability can be determined at a specific point in time ().The distribution (79) Similarly we can also obtain the distribution function, reliability function, and hazard rate of other theorems too.

Conclusions
In this paper the authors conclude the following.(i) Weibull distribution is a special case of -Generalized Exponential Distribution.Further if  tends to 0, then our lastly proved more general result leads to Weibull distribution.
(ii) Also if  tends to 1, then our newly introduced 3parameter extension of Generalized Exponential Distribution (GED) reduced to classical Exponential Distribution.
(iii) The moment generating functions (MGFs) obtained in this paper generalize the classical moment generating functions (MGFs) of the given distributions.