 # A weighted transmuted exponential distribution with environmental applications

In this paper, we introduce a new three-parameter distribution based on the combination of re-parametrization of the so-called EGNB2 and transmuted exponential distributions. This combination aims to modify the transmuted exponential distribution via the incorporation of an additional parameter, mainly adding a high degree of flexibility on the mode and impacting the skewness and kurtosis of the tail. We explore some mathematical properties of this distribution including the hazard rate function, moments, the moment generating function, the quantile function, various entropy measures and (reversed) residual life functions. A statistical study investigates estimation of the parameters using the method of maximum likelihood. The distribution along with other existing distributions are fitted to two environmental data sets and its superior performance is assessed by using some goodness-of-fit tests. As a result, some environmental measures associated with these data are obtained such as the return level and mean deviation about this level.

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## 1 Introduction

The precise analysis of a wide variety of data sets is limited by the use of models based on the classical distributions (normal, exponential, logistic…). For instance, the analysis of environmental data sets collecting from observations of complex natural phenomena needs special treatments to reveal all the underlying informations. Over the last decades, numerous solutions have been provided by the statisticians, including the elaboration of several methods which aim to increase the flexibility of the former classical distributions. Among these methods, a popular one that aims to construct a generator of distributions by compounding continuous distributions with well-known discrete distributions. This compounding is always motivated by practical problems as those involving cdf of minimum or maximum of several independent and identically random variables. An exhaustive survey on the construction of such generators, with the presentation of new ones, can be found in

, and the references therein. Among the long list, let us briefly present the EGNB2 distribution introduced by [22, Remark 2 (ii)]

. Using a cumulative distribution function (cdf)

, the general form of the associated cdf is given by

 FEGNB2(x)=[1+ηυG(x)α]−1η−1(1+ηυ)−1η−1. (1)

The EGNB2 distribution can be viewed as an extension of the G-negative binomial families introduced by  and . It enjoys remarkable theoretical and practical properties.

In this study, we consider a particular case of this EGNB2 distribution consisting in a re-parametrization for the parameters , and appearing in (1) as described below. Let , , and . That yields a cdf of the (simple) form:

 F(x)=[1+γG(x)]1γ+1−1(1+γ)1γ+1−1. (2)

Let us now explain the importance of this re-parametrization of (1), with some statistical features. One can observe that as the following integral form: , where denotes the pdf: . So it reveals to be a new particular case of the T-X family cdf introduced by . Another remark is that, when , we have and when , we have . This transformation of cdf corresponds to the one proposed in 

. All the resulting distributions have demonstrated nice properties in terms of analysis of real life data sets. Furthermore, let us observe that the probability density function (pdf) associated to (

2) is given by

 f(x)=(γ+1)[1+γG(x)]1γg(x)(1+γ)1γ+1−1.

Note that we can also express it as a weighted pdf: , where is a weight function and is a normalizing constant. It thus belongs to the family of weighted distributions. Further details on such family of distributions can be found in . On the other side,  introduced the transmuted exponential distribution defined by the following cdf: , , where denotes the cdf of the exponential distribution. Then, it is proved that the additional parameter can significantly increase the flexibility of the former exponential distribution, demonstrating a superiority in terms of fit in comparison to the former exponential distribution. We may refer the reader to , and the references therein.

In this paper, we introduce a new three-parameter distribution which combines the features of the distribution characterized by (2) and the transmuted exponential distribution. This combination aims to modify the former transmuted exponential distribution by incorporating the parameter

and takes benefit of the flexibility of the EGNB2 distribution. Its main role is to add a high degree of flexibility on the mode, and the skewness and kurtosis of the tail. We thus obtain a very flexible distribution, which opens new perspectives in terms of the construction of statistical models for data analysis. The theoretical and practical aspects are explored in an exhaustive way. The theoretical ones include expansions of the cdf, pdf, hazard rate function (hrf), quantile function, moments, moment generating function, various entropy measures, residual life functions, conditional moments, mean deviations and reversed residual life function. We investigate the estimation of its parameters via the maximum likelihood method. Two real-life data sets in environmental sciences are analyzed to show its superior performance in terms of fit in comparison to well-known distributions: The gamma distribution, the Marshal-Olkin exponential distribution

, the Nadarajah-Haghighi exponential distribution , the exponentiated exponential distribution , the transmuted Weibull distribution , the transmuted generalized exponential distribution , the transmuted linear exponential distribution  and the Kappa distribution . The best performance of the proposed distribution recommends it as a hydrologic probability model, such as the most known distributions: Kappa and gamma distributions. This motivates to estimate important hydrologic parameters of those data sets by making use of the distribution.

The rest of this article is organized as follows. In Section 2, we present our main distribution. Some of its mathematical properties are studied in Section 3. Residual life functions are determined in Section 4. Estimations of the parameters are investigated in Section 5. Applications to two real-life data sets are provided in Section 6. Concluding remarks are addressed in Section 7.

## 2 A new weighted transmuted exponential distribution

In this section, we precise what is the considered cdf given by (2).  and  introduced the quadratic rank transmutation map (QRTM) to propose a new distribution based on the Weibull/exponential one with great flexibility and nice fit for real-life data. In the current studies, it remains a serious competitor in terms of precision in modelling (see ). For these reasons, we use it in our study. We consider the cdf:

 G(x)=(θ+1)H(x)−θ[H(x)]2,θ∈[−1,1],

where is considered to be the cdf of the exponential distribution of parameter :

 G(x)=(θ+1)(1−e−λx)−θ(1−e−λx)2,x,λ>0.

Set the above expression into (2), we introduce a new cdf defined by

 F(x) =[1+γ(θ+1)(1−e−λx)−γθ(1−e−λx)2]1γ+1−1(1+γ)1γ+1−1 =[1+γ−γe−λx(1−θ+θe−λx)]1γ+1−1(1+γ)1γ+1−1,x>0,λ,γ>0.

Another useful expression is the following one:

 F(x)=(1+γ)1γ+1[1−γ1+γe−λx(1−θ+θe−λx)]1γ+1−1(1+γ)1γ+1−1. (3)

We will refer to the distribution given by (3) as the new weighted transmuted exponential and denote it by NWTE(, , ) with the considered parameters.

The corresponding pdf is given by

 (4)

The associated hrf is given by

 h(x)=λ[1−γ1+γe−λx(1−θ+θe−λx)]1γe−λx(1−θ+2θe−λx)1−[1−γ1+γe−λx(1−θ+θe−λx)]1γ+1. (5)

Let us now discuss the possible shapes of pdf (4) and hrf (5) as follows.

 limx→0f(x)=λ(1+θ)(1+γ)(1+γ)1γ+1−1,limx→+∞f(x)=0.

On the other side, we have

 limx→0h(x)=λ(1+θ)(1+γ)(1+γ)1γ+1−1,limx→+∞h(x)=λ.

In order to visualize the wide variety of shapes, some plots of the pdf (4) and hrf (5) are given in Figures 1 and 2. We see that has a great impact on the mode of the NWTE distribution. Moreover, the hrf also exhibits sudden spikes at the end of upside-down bathtub shapes, which manages the model to analyze a non-stationary real-life data.

## 3 Structural properties of the NWTE distribution

### 3.1 Expansion for the associated functions

Expansion for the cdf function. First of all, set , , . Note that we have , so is increasing. Since and , we have for all . Since and , the generalized binomial expansion, we have

 [1−γ1+γe−λx(1−θ+θe−λx)]1γ+1=+∞∑i=0(1/γ+1i)(−γ1+γ)ie−λix(1−θ+θe−λx)i (6) = +∞∑i=0i∑k=0(1/γ+1i)(ik)(−γ1+γ)ie−λix(1−θ)i−kθke−λkx=+∞∑i=0i∑k=0Hi,ke−λx(i+k),

where

 Hi,k=(1/γ+1i)(ik)(γθ−γ1+γ)i(θ1−θ)k.

Therefore we can expand the cdf function as

 (7)

Expansion for the pdf function. Similar mathematical arguments used for (6) give

 [1−γ1+γe−λx(1−θ+θe−λx)]1γ=+∞∑i=0i∑k=0Ai,ke−λx(i+k),

where

 Ai,k=(1/γi)(ik)(γθ−γ1+γ)i(θ1−θ)k.

Therefore

 f(x) = λ(1+γ)1γ+1(1+γ)1γ+1−1+∞∑i=0i∑k=0Ai,ke−λx(i+k)e−λx(1−θ+2θe−λx) (8) = +∞∑i=0i∑k=0Bi,k[(1−θ)e−λx(i+k+1)+2θe−λx(i+k+2)],

where

 Bi,k=λ(1+γ)1γ+1(1+γ)1γ+1−1Ai,k.

On the survival function. Note that

 S(x)=1−F(x)=(1+γ)1γ+1[1−{1−γ1+γe−λx(1−θ+θe−λx)}1γ+1](1+γ)1γ+1−1. (9)

Using (7), we have the following expansion

 (10)

Expansion for the hrf function. Using (5), (8) and (10), an expansion of the hrf function is given by

 (11)

Another expansion comes from the geometric series decomposition:

 11−[1−γ1+γe−λx(1−θ+θe−λx)]1γ+1=+∞∑m=0[1−γ1+γe−λx(1−θ+θe−λx)]m(1γ+1).

By (5) and similar mathematical arguments used for (6) give:

 h(x) =+∞∑m=0+∞∑i=0i∑k=0Gi,k,m[(1−θ)e−λx(i+k+1)+2θe−λx(i+k+2)],

where

 Gi,k,m=λ(m(1γ+1)+1γi)(ik)(γθ−γ1+γ)i(θ1−θ)k.

### 3.2 Quantile function

The quantile functions are in widespread use in general statistics to obtain mathematical properties of a distribution and often find representations in terms of lookup tables for key percentiles. For generating data from the NWTE model, let . Then, by inverting the cdf (3) and after some algebra, we get the quantile function

 Q(u)=1λ⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣−log⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩1−⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1+θ− ⎷(1+θ)2−4θγ⎡⎣{u([1+γ]1γ+1−1)+1}γγ+1−1⎤⎦2θ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦. (12)

The analysis of the variability of the skewness and kurtosis of X can be investigated based on quantile measures. The Bowley skewness is given by

 S=Q(3/4)−2Q(1/2)+Q(1/4)Q(3/4)−Q(1/4)

and the Moors’ kurtosis by

 K={Q(7/8)−Q(5/8)}+{Q(3/8)−Q(1/8)}Q(6/8)−Q(2/8),

where is given by (12).

These measures are less sensitive to outliers and they exist even for distributions without moments. Figure

3 displays plots of S and K as functions of and , which show their variability in terms of the shape parameters. Figure 3: Plots of the skewness and kurtosis of the NWTE distribution for λ=0.5.

### 3.3 Moments and moment generating function

Moments. Using equation (8) and the gamma function , the -th moments about the origin is given by

 μr =E(Xr)=∫+∞−∞xrf(x)dx =+∞∑i=0i∑k=0Bi,k[(1−θ)∫+∞0xre−λx(i+k+1)dx+2θ∫+∞0xre−λx(i+k+2)dx] =γ(r+1)λr+1+∞∑i=0i∑k=0Bi,k[1−θ(i+k+1)r+1+2θ(i+k+2)r+1]. (13)

The moment generating function. Similarly the moment generating function associated to the NWTE distribution is given by, for ,

 MX(t) =E(etx)=∫+∞−∞etxf(x)dx =+∞∑i=0i∑k=0Bi,k[(1−θ)∫+∞0etxe−λx(i+k+1)dx+2θ∫+∞0etxe−λx(i+k+2)dx] =+∞∑i=0i∑k=0Bi,k[1−θλ(i+k+1)−t+2θλ(i+k+2)−t]. (14)

### 3.4 Entropies

An entropy can be considered as a measure of uncertainty of probability distribution of a random variable. Therefore, we obtain three entropies for the NWTE distribution with investigating a numerical study among them.

Entropy 1. Let us consider the Shannon entropy : . One can observe that

 H(f) =−log⎡⎢⎣λ(1+γ)1γ+1(1+γ)1γ+1−1⎤⎥⎦−1γ∫+∞0f(x)log[1−γ1+γe−λx(1−θ+θe−λx)]dx +λE(X)−∫+∞0f(x)log[1−θ+2θe−λx]dx. (15)

Let us now expand the two integrals by using the logarithmic expansion: , . Since , we have

 ∫+∞0f(x)log[1−γ1+γe−λx(1−θ+θe−λx)]dx = −+∞∑m=11m(γ1+γ)m∫+∞0f(x)e−λmx(1−θ+θe−λx)mdx = −+∞∑m=1m∑ℓ=01m(mℓ)(1−θ)m−ℓ(γ1+γ)mθℓ∫+∞0f(x)e−λ(ℓ+m)xdx=+∞∑m=1m∑ℓ=0Rm,ℓ,

where

 Rm,ℓ=−1m(mℓ)(1−θ)m−ℓ(γ1+γ)mθℓMX[−λ(ℓ+m)],

denotes the moment generating function defined by (3.3).

For the second integral in (3.4), since , we have

 ∫+∞0f(x)log[1−θ+2θe−λx]dx=−+∞∑m=1θmm∫+∞0f(x)(1−2e−λx)mdx = −+∞∑m=1m∑ℓ=01m(mℓ)θm(−2)ℓ∫+∞0f(x)e−λℓxdx=+∞∑m=1m∑ℓ=0Um,ℓ,

where

 Um,ℓ=−1m(mℓ)θm(−2)ℓMX(−λℓ).

Entropy 2. Let us now focus our attention on the Rényi entropy : , with and . Similar mathematical arguments used for (6) give :

 [1−γ1+γe−λx(1−θ+θe−λx)]βγ=+∞∑i=0i∑k=0Ci,ke−λx(i+k),

where

 Ci,k=(β/γi)(ik)(γθ−γ1+γ)i(θ1−θ)k.

On the other side, observing that , similar mathematical arguments used for (6) give :

 (1−θ+2θe−λx)β=[1−θ(1−2e−λx)]β=+∞∑j=0j∑ℓ=0Dj,ℓe−λℓx,

where

 Dj,ℓ=(βj)(jℓ)(−θ)j(−2)ℓ.

Hence can be expanded as

 [f(x)]β=+∞∑i=0i∑k=0+∞∑j=0j∑ℓ=0Fi,k,j,ℓe−λx(i+k+ℓ+β),

where

 Fi,k,j,ℓ=⎡⎢⎣λ(1+γ)(1+γ)1γ+1−1⎤⎥⎦βCi,kDj,ℓ.

Hence

 ∫+∞−∞[f(x)]βdx=∫+∞0+∞∑i=0i∑k=0+∞∑j=0j∑ℓ=0Fi,k,j,ℓe−λx(i+k+ℓ+β)dx=1λ+∞∑i=0i∑k=0+∞∑j=0j∑ℓ=0Fi,k,j,ℓ1i+k+ℓ+β.

Therefore

 JR(β)=11−βlog(∫+∞−∞[f(x)]βdx)=11−β[−log(λ)+log(+∞∑i=0i∑k=0+∞∑j=0j∑ℓ=0Fi,k,j,ℓ1i+k+ℓ+β)].

Entropy 3. We now focus our attention on the entropy introduced by : , with and . Proceeding as for with instead of , we obtain

 [f(x)]2−δ=+∞∑i=0i∑k=0+∞∑j=0j∑ℓ=0Gi,k,j,ℓe−λx(i+k+ℓ+2−δ),

where

 Gi,k,j,ℓ=⎡⎢⎣λ(1+γ)(1+γ)1γ+1−1⎤⎥⎦2−δ((2−δ)/γi)(ik)(γθ−γ1+γ)i(θ1−θ)k(2−δj)(jℓ)(−θ)j(−2)ℓ.

Hence

 JMH(δ) = 1δ−1(∫+∞−∞[f(x)]2−δdx−1) = 1δ−1(1λ+∞∑i=0i∑k=0+∞∑j=0j∑ℓ=0Gi,k,j,ℓ1i+k+ℓ+2−δ−1).

Some numerical values for the three entropies are given in Table 1. It can be observed that these entropies decrease with increasing the parameter values. Moreover, one can see that has the smallest values comparing with the other entropies considered here.

### 3.5 Conditional moments and mean deviations

Here, we introduce an important lemma which will be used in the next sections.

###### Lemma 1.

Let and be the lower incomplete gamma function. Then we have

 Jr(t)=1λr+1+∞∑i=0i∑k=0Bi,k[(1−θ)γ{λ(i+k+1)t,r+1}(i+k+1)r+1+2θγ{λ(i+k+2)t,r+1}(i+k+2)r+1]. (16)
###### Proof.

Using the equation (8), we have

 Jr(t) =∫t0xrf(x)dx=+∞∑i=0i∑k=0Bi,k[(1−θ)∫t0xre−λx(i+k+1)dx+2θ∫t0xre−λx(i+k+2)dx] =1λr+1+∞∑i=0i∑k=0Bi,k[(1−θ)γ{λ(i+k+1)t,r+1}(i+k+1)r+1+2θγ{λ(i+k+2)t,r+1}(i+k+2)r+1].

The -th conditional moments of the NWTE distribution is given by

 E(Xr∣X>t)=11−F(t)∫+∞txrf(x)dx=1S(t)[E(Xr)−Jr(t)]. (17)

It can be expressed using (5), (3.3) and Lemma 1. The same remark holds for the -th reversed moments of the NWTE distribution given by

 E(Xr∣X≤t)=1F(t)∫t0xrf(x)dx=1F(t)Jr(t).

The mean deviations of about the mean can be expressed as and the mean deviations of about the median has the form .

## 4 (Reversed) Residual life functions

The residual life is described by the conditional random variable , . Using (10), the survival function of the residual lifetime for the NWTE distribution is given by

The associated cdf is given by

 FR(t)(x)=[1−γ1+γe−λ(x+t){1−θ+θe−λ(x+t)}]1γ+1−[1−γ1+γe−λt(1−θ+θe−λt)]1γ+11−[1−γ1+γe−λt(1−θ+θe−λt)]1γ+1.

The corresponding pdf is given by

 fR(t)(x)=λ[1−γ1+γe−λ(x+t){1−θ+θe−λ(x+t)}]1γe−λ(x+t)[1−θ+2θe−λ(x+t)]1−[1−γ1+γe−λt(1−θ+θe−λt)]1γ+1.

The associated hrf is given by

 hR(t)(x)=λ[1−γ1+γe−λ(x+t){1−θ+θe−λ(x+t)}]1γe−λ(x+t)[1−θ+2θe−λ(x+t)]1−[1−γ1+γe−λ(x+t){1−θ+θe−λ(x+t)}]1γ+1.

The mean residual life is defined as

 K(t)=E(R(t))=1S(t)∫+∞txf(x)dx−t=1S(t)[E(X)−J1(t)]−t,

where is given by (4), is mentioned in (9), is given by (3.3) and is stated in Lemma 1.

Further, the variance residual life is given by

 V(t) =Var(R(t))=2S(t)∫+∞txS(x)dx−2tK(t)−[K(t)]2 =1S(t)[E(X2)−J2(t)]−t2−2tK(t)−[K(t)]2,

where is given by (3.3) and is given by Lemma 1. Some numerical values for the mean residual life are displayed in Table 2 for various choices of the parameters and at the time points It can be seen that, the mean residual life increases with increasing the time points t, also decreases with increasing and .

### 4.2 Reversed residual life function

The reverse residual life is described by the conditional random variable , . Using (3), the survival function of the reversed residual lifetime for the NWTE distribution is given by

 S¯¯¯¯R(t)(x)=F(t−x)F(t)=[1−γ1+γe