JPS Journal of Probability and Statistics 1687-9538 1687-952X Hindawi 10.1155/2017/2106748 2106748 Research Article New Generalizations of Exponential Distribution with Applications Rather N. A. 1 http://orcid.org/0000-0002-0905-4684 Rather T. A. 1 Li Chin-Shang Department of Mathematics University of Kashmir Srinagar 190006 India kashmiruniversity.net 2017 672017 2017 05 12 2016 26 03 2017 07 06 2017 672017 2017 Copyright © 2017 N. A. Rather and T. A. Rather. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The main purpose of this paper is to present k-Generalized Exponential Distribution which among other things includes Generalized Exponential and Weibull Distributions as special cases. Besides, we also obtain three-parameter extension of Generalized Exponential Distribution. We shall also discuss moment generating functions (MGFs) of these newly introduced distributions.

1. Introduction

The gamma family distributions was discussed by Karl Pearson in 1895 as pointed out in Balakrishnan and Basu . 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  as Pearson’s Type X distribution. Steffensen , Teissier , and Weibull  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 <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi>K</mml:mi></mml:mrow></mml:math></inline-formula>-Generalized Exponential Distributions

We begin with some definitions which provide a base for the definition of K-Generalized Exponential Distributions.

The Euler gamma function Γ(α) is defined by the integral(1)Γα=0tα-1e-tdt,Rα>0.A random variable X is said to have gamma distribution with parameter α>0, if its p.d.f. is given by (2)fx=1Γαxα-1e-x,0<x<fx=0,elsewhere.Replacing x by x/λ, we get the following form of gamma distribution with parameters α,λ with α>0 and λ>0:(3)fx=1λαΓαxα-1e-x/λ,0<x<,α>0,λ>0.The gamma distribution with parameters α, λ 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 m and n is defined by (4)Bm,n=01xm-11-xn-1dx,Rem>0,Ren>0.If we take x=1-y, then, 0x1 implies 0y1, and we get (5)Bm,n=011-ym-1yn-1dy=01yn-11-ym-1dy=01xn-11-xm-1dx=Bn,m.In the literature (for reference see ), it is known that (6)Bn,m=ΓmΓnΓm+n.

In (5), if we take(7)x=1z+1then 0x1 implies 0z<, and we get (8)Bn,m=01z+1n-1zz+1m-1-1z+12dz=0zm-1z+1m+ndz.A continuous random variable X is said to have beta distribution with parameters m and n, if its p.d.f. is given by (9)fx=1Bm,nxm-11-xn-1,0x1fx=0,elsewhere.This distribution is known as beta distribution of 1st kind (for reference see ).

The beta distribution has an application to model a random phenomenon whose set of possible values is a finite interval a,b, which by letting a denote the origin and taking b-a as a unit measurement can be transformed into the interval 0,1.

A continuous random variable X is said to have beta distribution of 2nd kind with parameters m and n, if its p.d.f. is given by (10)fx=1Bm,nzm-11+zm+n,0z<,m,n>0,fx=0,elsewhere.More recently, Rahman et al.  (for more details see ) have defined k-gamma and k-beta distributions and their MGFs as follows.

For k>0 and z, the k-gamma function is defined by the integral (11)Γkz=0tz-1e-tk/kdt.For R(m)>0, R(n)>0, the k-beta function of two variables m and n is defined by (12)Bkm,n=1k01tm/k-11-tn/k-1dt.It is implicit in the literature (for reference see ) that (13)Bkm,n=ΓkmΓknΓkm+n.For the sake of completeness, we present a very simple proof of the relation (13).

We have (14)Bkm,n=1k01xm/k-11-xn/k-1dx.Put(15)x=cos2θ;then(16)dx=-2cosθsinθdθ,and we get (17)Bkm,n=2k0π/2cosθ2m/k-1sinθ2n/k-1dθ.Now (18)Γkm=0tm-1e-tk/kdt.Put(19)tkk=x2or (20)t=kx21/k,so that (21)dt=2kk1/kx2/k-1dx=2k1/k-1x2/k-1dx;we get (22)Γkm=2k1/k-10km-1/kx2/km-1e-x2x2/k-1dx=2km/k-10x2m/k-1e-x2dx.Since the integrals involved are convergent, we have(23)ΓkmΓkn=2km/k-10x2m/k-1e-x2dx2kn/k-10y2n/k-1e-y2dy=4km+n/k-200x2m/k-1y2n/k-1e-x2+y2dxdy.Put (24)x=rcosθ,y=rsinθ,so that (25)dxdy=rdrdθ;we get (26)ΓkmΓkn=4km+n/k-200π/2cosθ2m/k-1sinθ2n/k-1e-r2r2m/k+2n/k-1drdθ=2k0π/2cosθ2m/k-1sinθ2n/k-1dθ2km+n/k-10r2m+n/k-1e-r2dr.Using (17) and (22), we get (27)ΓkmΓkn=Bkm,nΓkm+n,which gives (28)Bkm,n=ΓkmΓknΓkm+n.This completes the proof of the relation (13).

Corollary 1.

One has the following:

Γkk=1.

Bkk/2,k/2=π/k.

Γkk/2=π/k.

Proof.

Put m=k in (22); we get (29)Γkk=20xe-x2dx=-e-x20=1.From (13), we have (30)Bkm,n=ΓkmΓknΓkm+n.We take m=n=k/2; we get(31)Bkk2,k2=Γkk/2Γkk/2Γkk=Γkk2Γkk2=Γkk22.This gives (32)Γkk2=Bkk2,k2.Putting m=k/2 and n=k/2 in (17), we get (33)Bkk2,k2=2k0π/2dθ=2kθ0π/2=2kπ2=πk,which establishes (ii).

Putting the value of Bkk/2,k/2 in (32), we get (34)Γkk2=πk,which proves (iii).

A random variable X of continuous type is said to have Weibull distribution if its probability density function is given by(35)fx,α,β,ν=βαx-ναβ-1e-x-ν/αβ,x>νfx,α,β,ν=0,elsewhere,xν.

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  (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 , Gupta and Kundu , and Gupta et al. .

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.

3. Generalized Exponential Distribution (GED)

A continuous random variable X whose p.d.f. is given by (36)fx=βe-βxifx00ifx<0,β>0,is said to have an Exponential Distribution.

The Generalized Exponential Distribution (GED) introduced by Mubeen et al.  has p.d.f.(37)fx,α,β=αβ1-e-βxα-1e-βx,x>0  for  α,β>0fx,α,β=0,otherwise.The MGF of (37) is given by (38)MXt=EetX=0etxαβ1-e-βxα-1e-βxdx=αBα,1-tβ,where B(m,n)=01yn-1(1-y)m-1dy.

Replacing β by 1/λ and x by (x-μ) in (37), we get the following form of Generalized Exponential Distribution (GED):(39)fx,α,λ,μ=αλ1-e-x-μ/λα-1e-x-μ/λ,x>μ,α>0,λ>0.

4. 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 k>0 and call it k-Generalized Exponential Distribution (k-GED). In fact, we prove the following result, which included Generalized Exponential Distribution as a special case.

Theorem 2.

Let X be a random variable of continuous type and let α>0,β>0, and k>0 be the parameters; then the function (40)fx,α,β,k=αβ1-e-βxk/kα-1xk-1e-βxk/k,x>0fx,α,β,k=0,elsewhereis the p.d.f. of random variable X of continuous type.

Remark 3.

If we take k=1, it reduces to Generalized Exponential Distribution.

Proof of Theorem <xref ref-type="statement" rid="thm1">2</xref>.

Clearly(41)fx,α,β,k0x>0,α>0,β>0,k>0.Now (42)0fx,α,β,kdx=0αβ1-e-βxk/kα-1xk-1e-βxk/kdx=α01-e-βxk/kα-1βxk-1e-βxk/kdx=α1-e-βxk/kαα0=1-e-βxk/kα0=1-0=1.Hence f(x,α,β,k) is a p.d.f. of random variable X of continuous type.

5. The Moment Generating Function (MGF) of Theorem <xref ref-type="statement" rid="thm1">2</xref>

In this section, we derive MGF of the random variable X having k-Generalized Exponential Distribution in terms of new parameter k>0; we have (43)Mkt=EetXk=0etxkfx,α,β,kdx=kαβ0etxk1-e-βxk/kα-1xk-1e-βxk/kdx.Put (44)e-βxk/k=y;then(45)e-βxk/k-βxk-1dx=dy,exk=y-k/βso that (46)etxk=y-tk/β.Therefore, (47)Mkt=α01y-tk/β1-yα-1dy=α01y1-tk/β-11-yα-1dy=α011-yα-1y1-tk/β-1dy=αBα,1-tkβ=αΓαΓ1-tk/βΓα+1-tk/β=Γα+1Γ1-tk/βΓα+1-tk/β.Our next theorem is also a generalization of Exponential Distribution in terms of new variable k, which includes Weibull distribution as a special case.

Theorem 4.

Let X be a random variable of continuous type and let α>0,  β>0, and k>0 be the parameters; then the function (48)fx,α,β,k=kαβ1-e-βxkα-1xk-1e-βxk,0<x<fx,α,β,k=0,elsewhere,is the p.d.f. of random variable X of continuous type.

Remark 5.

For k=1, K-Generalized Exponential Theorem 4 reduces to Classical Exponential Distribution.

Proof of Theorem <xref ref-type="statement" rid="thm2">4</xref>.

Clearly (49)fx,α,β,k0x>0,α>0,β>0,k>0.Now (50)0fx,α,β,kdx=kαβ01-e-βxkα-1xk-1e-βxkdx=α01-e-βxkα-1kβxk-1e-βxkdx=α1-e-βxkαα0=1-e-βxkα0=1-0=1.Hence f(x,α,β,k) is a p.d.f. of random variable X of continuous type.

Remark 6 (Weibull distribution is a special case of Theorem <xref ref-type="statement" rid="thm2">4</xref>).

To see this, we take α=1,β=1 in Theorem 4; it follows that (51)fx,k=kxk-1e-xk=e-xkddxxk,x>0fx,k=0,x0.Replacing x by x-μ/α, we get(52)fx,α,k,ν=e-x-ν/αkddxx-ναk,x-να>0fx,α,k,ν=0,x-να0or(53)fx,α,k,ν=kαx-ναk-1e-x-ν/αk,x>ν,fx,α,k,ν=0,xν,is a p.d.f. of X, which is clearly the density of Weibull Distribution.

6. The Moment Generating Function (MGF) of Theorem <xref ref-type="statement" rid="thm2">4</xref>

In this section, we derive MGF of the random variable X having k-Generalized Exponential Distribution in terms of a new parameter k>0; we have (54)Mkt=EetXk=0etxkfx,α,β,kdx=kαβ0etxk1-e-βxkα-1xk-1e-βxkdx.Put e-βxk=y; then(55)exk=y-1/β,e-βxk-kβxk-1dx=dy.Therefore, (56)Mkt=α01y-t/β1-yα-1dy=α01y1-t/β-11-yα-1dy=α011-yα-1y1-t/β-1dy=αBα,1-tβ=αΓαΓ1-t/βΓα+1-t/β=Γα+1Γ1-t/βΓα+1-t/β.

We also present the following three-parameter extension of Generalized Exponential Distribution (GED). In fact, we prove the following.

Theorem 7.

Let X be a continuous random variable; then the function(57)fx,α,β,δ=αβδ1-1-δα1-δe-βxα-1e-βx,x>0,α>0,β>0,0<δ1,fx,α,β,δ=0,otherwiseis the p.d.f. of random variable X of continuous type.

Proof of Theorem <xref ref-type="statement" rid="thm3">7</xref>.

Clearly f(x,α,β,δ)0 for all x>0,α>0,β>0,δ>0.

Now (58)0fx,α,β,δdx=αβδ1-1-δα01-δe-βxα-1e-βxdx=α1-1-δα01-δe-βxα-1βδe-βxdx=α1-1-δα1-δe-βxαα0=1-1-δα1-1-δα=1.This shows thatf(x) is a p.d.f. of the random variable X. This proves Theorem 7.

If we replace β by 1/λ and x by (x-μ) in Theorem 7, we get the following form of Generalized Exponential Distribution (GED):(59)fx,α,λ,μ,δ=αδλ1-1-δα1-δe-x-μ/λα-1e-x-μ/λ,x>μ,α>0,λ>0,μ>0,0<δ1.fx,α,λ,μ,δ=0,otherwise.

Remark 8.

For δ=1, Theorem 7 reduced to Generalized Exponential Distribution (GED).

Remark 9.

Taking δ=1 in (59), we get relation (39).

7. The Moment Generating Function (MGF) of Theorem <xref ref-type="statement" rid="thm3">7</xref>

(60) M X t = E e t X = 0 e t x f x , α , β , δ d x = α β δ 1 - 1 - δ α 0 e t x 1 - δ e - β x α - 1 e - β x d x = α δ 1 - 1 - δ α 0 1 - δ e - β x α - 1 e t x β e - β x d x ,

Put (61)e-βx=y,so that (62)-βe-βxdx=dy,etx=y-t/β;we get(63)Mt=αδ1-1-δα011-δyα-1y-t/βdy,0<δ1.Mt=α1-1-δαBδα,1-tβ,where (64)Bδ=Bδm,n=011-δym-1yn-1dy,m>0,n>0,0<δ1.

Remark 10.

For δ=1, we have (65)B1m,n=011-ym-1yn-1dy=Bm,n.And (63) reduces to Remark 8.

Remark 11.

If in Bδm,n=011-δym-1yn-1dy, we put x=δy;

then (66)dy=1δdxand we get(67)Bδm,n=0δ1-xm-1xδn-11δdx=1δn0δ1-xm-1xn-1dx,0δ1.

Remark 12.

Letting δ0 in (57) and noting that(68)Ltδ0·αδ1-1-δα=1,we get(69)fx,β=βe-βx,ifx0,β>0,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.

Theorem 13.

Let X be a random variable of continuous type. If δ>0,β>0, and k>0 are the parameters, then the function (70)fx,δ,β,k=kδβ1-1-δk1-δe-xβk-1xβ-1e-xβ,x>0,0<δ<1,fx,δ,β,k=0,elsewhere,x0,is the p.d.f. of the random variable X.

Proof of Theorem <xref ref-type="statement" rid="thm4">13</xref>.

Clearly (71)fx,δ,β,k0x>0,δ>0,β>0,k>0.Now (72)0fx,δ,β,kdx=kδβ01-δe-xβk-1xβ-1e-xβdx=k1-1-δk01-δe-xβk-1βδe-xβxβ-1dx=k1-1-δk1-δe-xβkk0=k1-1-δk1-1-δkk=1.This shows that f(x,α,β,k) is the p.d.f. of random variable X of continuous type. This proves Theorem 13.

Remark 14.

Weibull Distribution is the limiting case of Theorem 13. To see this, we let δ0 in Theorem 13 and note that (73)limδ0kδ1-1-δk=1,so that (74)fx,β=βe-xβxβ-1=e-xβddxxβ,x>0,fx,β=0,x0,is the p.d.f. of random variable X of continuous type.

Replacing x by x-ν/α, it follows that (75)fx,α,β,ν=βe-x-ν/αβddxx-ναβ,x-να>0fx,α,β,ν=0,x-να0.Equivalently (76)fx,α,β,ν=βαx-ναβ-1e-x-ν/αβ,x>ν,fx,α,β,ν=0,xν,is a p.d.f. of X, which is clearly density of Weibull Distribution.

8. Applications

The applications of Exponential Distribution have been widespread, which include models to determine bout criteria for analysis of animal behaviour ; design rainfall estimation in the Coast of Chiapas ; analysis of Los Angeles rainfall data ; software reliability growth models for vital quality metrics ; models for episode peak and duration for ecohydroclimatic applications ; estimating mean life of power system equipment with limited end-of-life failure data ; and cure rate modeling (Kannan et al. (2010). In related work, the closeness of the exponentiated Exponential Distribution with the Weibull, gamma, and log-normal distributions is studied in Gupta and Kundu (2003a), (2003b), (2004), (2006)) and . Some generalizations of the exponentiated Exponential Distribution are discussed in Nadarajah and Kotz .

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, λ, or mean time between failure (MTBF = 1/λ), the reliability can be determined at a specific point in time (t). The distribution function, reliability function, and hazard rate of Theorem 2 are given by(77)Fx,α,β,k=PXx=0xfx,α,β,xdx=1-e-βxk/kα,x>0,α>0,β>0.The survival function S(x,α,β,k) is given by(78)Sx,α,β,k=1-Fx,α,β,k=1-1-e-βxk/kα,x>0.The hazard rate h(x,α,β,k) is given by(79)Sx,α,β,k=Fx,α,β,kSx,α,β,k=1-e-βxk/kα-1xk-1e-βxk/k1-1-e-βxk/kα,x>0.Similarly we can also obtain the distribution function, reliability function, and hazard rate of other theorems too.

9. Conclusions

In this paper the authors conclude the following.

(i) Weibull distribution is a special case of k-Generalized Exponential Distribution. Further if δ tends to 0, then our lastly proved more general result leads to Weibull distribution.

(ii) Also if δ tends to 1, then our newly introduced 3-parameter 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.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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