**1. Introduction**

Analysis of lifetime data is an important subject in many fields, including reliability, social sciences, biomedical, engineering and other fields. In practice, it has been observed that many phenomena do not follow any of the classical distributions; for this reason, many efforts have been made in the last few decades to introduce new generators or families of distributions that extend these classical distributions to provide considerable flexibility in modeling data in diverse spectrums. Many authors have suggested new generators or families in the literature, for example, and not exclusively: Marshall and Olkin (1997) [1] introduced the Marshall–Olkin family, Gupta et al. (1998) [2] introduced the exponentiated-G family, Eugene et al. (2002) [3] proposed the beta-G family, Cordeiro and Castro (2011) [4] suggested the Kumaraswamy-G family, Alexander et al. (2012) [5] presented the McDonald-G family, Alzaatreh et al. (2013) [6] proposed the transformed-transformer (T-X) family, Bourguignon et al. (2014) [7] presented the Weibull-G family, Tahir et al. (2015) [8] studied the odd generalized exponential-G family, Cordeiro et al. (2016) [9] discussed the Zografos Balakrishnan odd log-logistic family, Gomes-Silva et al. (2017) [10] presented the odd Lindley-G family, Alizadeh et al. (2017) [11] provided the Gompertz-G family and Jamal et al. (2017) [12] defined the odd Burr-III family, among others. For a clearer understanding of the odds ratio to define new G-classes, we motivate the readers to Khan et al. (2021) [13], in which the authors adopted a unique odd function to propose an alternate generalized odd generalized exponential-G family.

**Citation:** Jamal, F.; Handique, L.; Ahmed, A.H.N.; Khan, S.; Shafiq, S.; Marzouk, W. The Generalized Odd Linear Exponential Family of Distributions with Applications to Reliability Theory. *Math. Comput. Appl.* **2022**, *27*, 55. https://doi.org /10.3390/mca27040055

Academic Editor: Sandra Ferreira

Received: 20 May 2022 Accepted: 21 June 2022 Published: 23 June 2022

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The linear exponential or (linear failure rate) distribution is the distribution of the minimum of two independent random variables *Z*<sup>1</sup> and *Z*<sup>2</sup> having exponential (*a*) and Rayleigh (*b*) (Sen and Bhattacharyya, 1995 [14]). Therefore, the variables have exponential and Rayleigh distributions as special cases, which are well-known distributions for modeling lifetime data in reliability and medical studies. The linear exponential distribution is used to model phenomena with linearly increasing failure rates, but it does not provide a reasonable fit for modeling phenomena with decreasing, non-linear increasing, or non-monotonic failure rates, which include the bathtub and upside-down bathtub, among others. These phenomena are common in reliability and biological studies. This motivated us to introduce generalizations of linear, exponential distribution so that their goodness of fit measures may improve the tail properties. Our motivations and the main goals of this paper are to propose a random variable that follows the linear exponential distribution as a new generator to introduce new models which can yield all types of the hazard rate functions with improved goodness of fit properties for real-life data.

#### **2. The Generalized Odd Linear Exponential (GOLE-F) Family**

Suppose the random variable *Z* has a linear exponential distribution with parameters *a*, *b* ≥ 0 where *a* + *b* > 0, then its cumulative distribution function (CDF) and probability density function (PDF) are, respectively,

$$R(z) = 1 - e^{-(az + \frac{b}{2}z^2)}, \; z \ge 0 \tag{1}$$

$$r(z) = (a+bz)e^{-\left(az+\frac{b}{2}z^2\right)}.z>0. \tag{2}$$

Adopting the T-X framework defined by the authors in [6], for any power parameter *c* > 0, we define the CDF of a new wider family called the generalized odd linear exponential ("GOLE-F" for short) family by

$$G(x;a,b,c,\Phi) = \int\_0^{\frac{F(x;\Phi)^c}{1-F(x;\Phi)^c}} (a+bz)e^{-\left(az+\frac{b}{2}z^2\right)} dz = 1 - \exp\left[-\left(\frac{aF(x;\Phi)^c}{1-F(x;\Phi)^c} + \frac{b}{2}\left(\frac{F(x;\Phi)^c}{1-F(x;\Phi)^c}\right)^2\right)\right],\tag{3}$$

where *W*[*F*(*x*)] = *<sup>F</sup>*(*x*;*φ*) *c* 1−*F*(*x*;*φ*) *<sup>c</sup>* is the link function with *F*(*x*; *φ*) as the baseline CDF of an absolutely continuous distribution with parameter vector *φ* and pdf *f*(*x*; *φ*).

The PDF of GOLE-F corresponding to the CDF in Equation (3) is provided by

$$\log \left( \mathbf{x}; a, b, c, \boldsymbol{\Phi} \right) = \left[ \frac{c f(\mathbf{x}; \boldsymbol{\Phi}) F(\mathbf{x}; \boldsymbol{\Phi})^{\varepsilon - 1} \left( a + (b - a) F(\mathbf{x}; \boldsymbol{\Phi})^{\varepsilon} \right)}{\left( 1 - F(\mathbf{x}; \boldsymbol{\Phi})^{\varepsilon} \right)^{3}} \right] \times \exp \left[ - \left( \frac{a F(\mathbf{x}; \boldsymbol{\Phi})^{\varepsilon}}{1 - F(\mathbf{x}; \boldsymbol{\Phi})^{\varepsilon}} + \frac{b}{2} \left( \frac{F(\mathbf{x}; \boldsymbol{\Phi})^{\varepsilon}}{1 - F(\mathbf{x}; \boldsymbol{\Phi})^{\varepsilon}} \right)^{2} \right) \right]. \tag{4}$$

Henceforth, for any parent model, we will simply write *F*(*x*) = *F*(*x*; *φ*) as the distribution function and *f*(*x*) = *f*(*x*; *φ*) as the density function. Further, any random variable *X* with density function (4) is denoted by *X* ∼ *GOLE* − *F* (*a*, *b*, *c*, *φ*).

The hazard rate function (HRF) and reversed hazard rate function (RHRF) of the random variable *X* are, respectively,

$$h(\mathbf{x}; a, b, c, \Phi) = \frac{\varepsilon f(\mathbf{x}) F(\mathbf{x})^{c-1} (a + (b - a)F(\mathbf{x})^c)}{\left(1 - F(\mathbf{x})^c\right)^3},\tag{5}$$

and

$$\pi(\mathbf{x};a,b,c,\Phi) = \frac{\left\{cf(\mathbf{x})F(\mathbf{x})^{c-1}(a+(b-a)F(\mathbf{x})^c)\right\}e^{-\left(\frac{aF(\mathbf{x})^{c'}}{1-F(\mathbf{x})^c} + \frac{b}{2}\left(\frac{F(\mathbf{x})^c}{1-F(\mathbf{x})^c}\right)^2\right)}}{\left(1-F(\mathbf{x})^c\right)^3\left\{1-e^{-\left(\frac{aF(\mathbf{x})^c}{1-F(\mathbf{x})^c} + \frac{b}{2}\left(\frac{F(\mathbf{x})^c}{1-F(\mathbf{x})^c}\right)^2\right)}}.\tag{6}$$

The quantile function of the random variable *X* can be obtained by inverting Equation (3), and hence the GOLE-F distribution can be simulated easily from the following Equation.

$$X = Q(lI) = F^{-1} \left( \left[ \frac{-a + \sqrt{a^2 - 2b \log(1 - lI)}}{b - a + \sqrt{a^2 - 2b \log(1 - lI)}} \right]^{1/c} \right) \tag{7}$$

where *U* has a uniform distribution over the interval (0,1), in particular, if *u* = 1/2 we obtain the median of the random variable *X* as follows:

$$M = Q\left(\frac{1}{2}\right) = F^{-1}\left(\left[\frac{-a + \sqrt{a^2 - 2b\log(1 - 1/2)}}{b - a + \sqrt{a^2 - 2b\log(1 - 1/2)}}\right]^{1/\varepsilon}\right). \tag{8}$$

#### **3. Special Model of the GOLE-F Family**

In this section, we provide two extended distributions as special models of the GOLE-F family and display their plots of density and hazard rate functions.

## *3.1. The* Generalized Odd Linear Exponential-Weibull *(GOLE-W) Distribution*

Consider the Weibull distribution with density and distribution functions *f*(*x*; *λ*, *β*) = *λβxβ*−1*e*−*λx<sup>β</sup>* and *<sup>F</sup>*(*x*; *<sup>λ</sup>*, *<sup>β</sup>*) <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*λx<sup>β</sup>* , respectively, where *λ*, *β* > 0 and *x* ≥ 0. Then, the GOLE-W distribution has (PDF) provided by

$$\begin{split} g(\mathbf{x};a,b,c,\lambda,\beta) &= \left[ \frac{c\lambda\mathfrak{f}\mathbf{x}^{\beta-1}\mathfrak{e}^{-\lambda\mathbf{x}^{\beta}} \left(1-\mathfrak{e}^{-\lambda\mathbf{x}^{\beta}}\right)^{c-1} \left(a+(b-a)\left(1-\mathfrak{e}^{-\lambda\mathbf{x}^{\beta}}\right)^{c}\right)}{\left(1-\left(1-\mathfrak{e}^{-\lambda\mathbf{x}^{\beta}}\right)^{c}\right)^{c}} \right] \\ &\times \exp\left[-\left(\frac{a\left(1-\mathfrak{e}^{-\lambda\mathbf{x}^{\beta}}\right)^{c}}{1-\left(1-\mathfrak{e}^{-\lambda\mathbf{x}^{\beta}}\right)^{c}}+\frac{b}{2}\left(\frac{\left(1-\mathfrak{e}^{-\lambda\mathbf{x}^{\beta}}\right)^{c}}{1-\left(1-\mathfrak{e}^{-\lambda\mathbf{x}^{\beta}}\right)^{c}}\right)^{2}\right)\right]. \end{split}$$

Figure 1a show a wealth of possible shapes of the distribution once different choices of the parameters are made. For example, the shape can be U and inverted-U, rightskewed, reversed-J shape or symmetrical. Additionally, Figure 1b reveal that the HRF of the GOLE-W distribution can be increasing–constant, constant–monotone–increasing or monotone–increasing shapes.

**Figure 1.** (**a**) Density function and (**b**) hazard rate plots of the GOLE-W distribution for different parameter values.

#### *3.2. The Generalized Odd Linear Exponential-Exponential (GOLE-E) Distribution*

Consider the Exponential distribution with density and distribution functions *f*(*x*; *λ*) = *<sup>λ</sup>e*−*λ<sup>x</sup>* and *<sup>F</sup>*(*x*; *<sup>λ</sup>*) <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*λx*, respectively, where *<sup>λ</sup>* <sup>&</sup>gt; 0 and *<sup>x</sup>* <sup>≥</sup> 0. Then, the GOLE-E distribution has (PDF) provided by

$$\operatorname{g}(x;a,b,c,\lambda) = \left[\frac{c\lambda e^{-\lambda x} \left(1 - e^{-\lambda x}\right)^{c-1} \left(a + (b-a)\left(1 - e^{-\lambda x}\right)^{c}\right)}{\left(1 - \left(1 - e^{-\lambda x}\right)^{c}\right)^{b}}\right] \times \exp\left[-\left(\frac{a\left(1 - e^{-\lambda x}\right)^{c}}{1 - \left(1 - e^{-\lambda x}\right)^{c}} + \frac{b}{2}\left(\frac{\left(1 - e^{-\lambda x}\right)^{c}}{1 - \left(1 - e^{-\lambda x}\right)^{c}}\right)^{2}\right)\right] \times \exp\left[-\left(\frac{a\left(1 - e^{-\lambda x}\right)^{c}}{1 - \left(1 - e^{-\lambda x}\right)^{c}} + \frac{b}{2}\left(\frac{\left(1 - e^{-\lambda x}\right)^{c}}{1 - \left(1 - e^{-\lambda x}\right)^{c}}\right)^{2}\right)\right]$$

Figure 2a show possible shapes of the GOLE-E distribution for different choices of the parameters. The shapes of pdf can be right-skewed, or symmetrical. Further, Figure 2b reveal that the HRF of the GOLE-E distribution can be decreasing–constant, monotone– increasing or bathtub shape. The PDF and HRF of the GOLE-W and GOLE-E distributions for some selected values of the parameters indicate the flexibility of the new family.

**Figure 2.** Plots of (**a**) density function and (**b**) hazard rate of the GOLE-E distribution for different parameter values.

#### **4. Mathematical Properties of the GOLE-F Family**

In this section, some mathematical properties of the GOLE-F family are obtained.

#### *4.1. Asymptotic Behavior of GOLE-F Family*

First of all, for the statements of the following results, we recall that *F*(*x*) is the CDF of an absolutely continuous distribution with pdf *f*(*x*).

**Proposition 1.** *The asymptotes corresponding to Equations (3)–(5) when x* → −∞ *are provided by*

$$G(\mathbf{x}) \sim a \, F(\mathbf{x})^c,\tag{9}$$

$$g(\mathbf{x}) \sim c \, a \, f(\mathbf{x}) F(\mathbf{x})^{c-1} \, \tag{10}$$

$$h(\mathbf{x}) \sim c \, a \, f(\mathbf{x}) F(\mathbf{x})^{c-1}. \tag{11}$$

**Proposition 2.** *The asymptotes corresponding to Equations (3)–(5) when x* → ∞ *are provided by*

$$1 - G(\mathbf{x}) \sim 1 - e^{-\left(\frac{a}{1 - F(\mathbf{x})^2} + \frac{b}{2}\left(\frac{1}{1 - F(\mathbf{x})^2}\right)^2\right)}\,,\tag{12}$$

$$g(\mathbf{x}) \sim \frac{bcf(\mathbf{x})}{\left(1 - F(\mathbf{x})^c\right)^3} \ e^{-\left(\frac{\mathbf{g}}{1 - F(\mathbf{x})^c} + \frac{\mathbf{k}}{2}\left(\frac{1}{1 - F(\mathbf{x})^c}\right)^2\right)}\tag{13}$$

$$h(\mathbf{x}) \sim \frac{bcf(\mathbf{x})}{\left(1 - F(\mathbf{x})^c\right)^3}.\tag{14}$$

*For detail see Appendix A*.

#### *4.2. Useful Expansions for CDF and PDF of the New Family*

Using the power series for the exponential function and the generalized binomial expansion

$$e^{-z} = \sum\_{i=0}^{\infty} \frac{(-1)^i z^i}{i!} \rho$$

and

$$(1 - v)^n = \sum\_{i=0}^{\infty} (-1)^i \binom{n}{i} v^i \lambda^i$$

respectively, where |*v*| < 1 and *n* is any real number, we can rewrite the CDF of the GOLE-F family as follows:

$$G(x;a,b,c,\Phi) = 1 - \sum\_{i=0}^{\infty} \sum\_{j=0}^{i} \sum\_{k=0}^{\infty} \frac{(-1)^{i} b^{j} a^{i} \binom{i}{j} \binom{i+j+k-1}{k}}{i! 2^{j}} F(x)^{c(i+j+k)}.\tag{15}$$

Again, based on the binomial expansion, we find

$$F(\mathbf{x})^{c(i+j+k)} = (1 - (1 - F(\mathbf{x}))^{c(i+j+k)} = \sum\_{m=0}^{\infty} \sum\_{l=m}^{\infty} (-1)^{l+m} \binom{l}{m} \binom{c(i+j+k)}{l} F(\mathbf{x})^m. \tag{16}$$

From (15) and (16), we obtain

$$G(\mathbf{x}; a, b, c, \Phi) = 1 - \sum\_{m=0}^{\infty} \omega\_m F(\mathbf{x})^m,\tag{17}$$

where

$$
\omega\_{\mathfrak{m}} = \sum\_{l=m}^{\infty} \sum\_{i=0}^{\infty} \sum\_{j=0}^{i} \sum\_{k=0}^{\infty} \rho\_{i,j,k,l}(a,b,c)\_{k}
$$

and

$$
\rho\_{i,j,k,l}(a,b,c) = \frac{(-1)^{i+l+m} b^j a^i \binom{i}{j} \binom{l}{m} \binom{i+j+k-1}{k} \binom{c(i+j+k)}{l}}{i! 2^j}.
$$

Now, we can write the CDF of the GOLE-F family in Equation (17), as

$$G(x;a,b,c,\Phi) = \sum\_{m=0}^{\infty} \delta\_m F(x)^m. \tag{18}$$

where *δ*<sup>0</sup> = 1 − *ω*0, and *δ<sup>m</sup>* = −*ω<sup>m</sup>* for *m* = 1, 2, ... By differentiating Equation (18), we obtain the expansion of the density function of the GOLE-F family as an infinite linear mixture of exp-F densities in the following form

$$\lg(\mathbf{x}; a, b, c, \Phi) = \sum\_{m=0}^{\infty} \delta\_{m+1} \pi\_{m+1}(\mathbf{x}),\tag{19}$$

where *πm*+1(*x*) = (*m* + 1)*f*(*x*)*F*(*x*)*<sup>m</sup>* is the exp-F density function with power parameter (*m* + 1). Now, if the random variable *Ym*+<sup>1</sup> has the density function *πm*+1(*x*), then many mathematical properties of the random variable *X*, including the ordinary and incomplete moments and moment generating function can easily be obtained based on the exp-F distribution.

#### *4.3. Moments*

Suppose that the random variable *Ym*+<sup>1</sup> has the density function *πm*+1(*x*) in (19), then the *n*th moment of the random variable *X* can be obtained from

$$\mu\_n' = E(X^n) = \sum\_{m=0}^{\infty} \delta\_{m+1} E\left(Y\_{m+1}^n\right). \tag{20}$$

A second alternative formula for *μ <sup>n</sup>* in terms of the baseline qf. *QF*(*u*) can be obtained as

$$
\mu\_n' = \sum\_{m=0}^{\infty} \delta\_{m+1}(m+1) \int\_0^1 Q\_F(u)^n u^m du,\tag{21}
$$

where *QF*(*u*) = *<sup>F</sup>*−1(*u*) is the qf of the parent distribution and *<sup>u</sup>* ∈ (0, 1).

The incomplete moments have an important role in measuring inequality, for example, income quantiles, the mean deviations and Lorenz and Bonferroni curves. The *n*th incomplete moment of *X* is provided by

$$\eta\_n(z) = \sum\_{m=0}^{\infty} \delta\_{m+1}(m+1) \int\_0^{\mathbb{F}(z)} Q\_{\mathbb{F}}(u)^n u^m du. \tag{22}$$

The last integral can be computed analytically or numerically for most baseline distributions. Bonferroni and Lorenz curves have applications in many different areas such as economics to study income and poverty, reliability, demography, insurance and medicine. For a random variable *X*, the Bonferroni and Lorenz curves are defined by *B*(*p*) = *η*1(*q*)/*pE*(*X*) and *L*(*p*) = *η*1(*q*)/*E*(*X*), respectively, where *p* is a given probability, *q* = *Q*(*p*) and *η*1(*q*) is the first incomplete moment that can be calculated from the above Equation with *r* = 1 at *q*. Table 1 display the mean, variance, skewness and kurtosis of the GOLE-E distribution for some choices values of the parameters. We note from Table 1 that the skewness of the GOLE-E distribution is always positive, whereas the kurtosis of the GOLE-E distribution varies only in the interval (1.0571, 2.6112).

**Table 1.** Mean, variance, skewness and kurtosis of the GOLE-E distribution with different values of *a*, *b*, *c* and *λ* = 1.


#### *4.4. Generating Function*

Here, we provide three formulae for the mgf *M*(*t*) = *E etX* of the random variable *X*. The first one is provided by

$$M(t) = \sum\_{n=0}^{\infty} \frac{t^n}{n!} \mu\_{n\prime}^{\prime} \tag{23}$$

where *μ <sup>n</sup>* is the *n*th moment of the random variable *X*. A second formula for *M*(*t*) comes from (19) as

$$M(t) = \sum\_{m=0}^{\infty} \delta\_{m+1} M\_{m+1}(t),\tag{24}$$

where *Mm*+1(*t*) is the mgf of the random variable *Ym*+<sup>1</sup> ∼exp-F (*m* + 1). A third formula for *M*(*t*) can also be derived based on (19) in terms of the baseline qf. *QF*(*u*) as

$$M(t) = \sum\_{m=0}^{\infty} \delta\_{m+1}(m+1) \int\_0^1 \exp(tQ\_F(u)) u^m du,\tag{25}$$

where *QF*(*u*) = *<sup>F</sup>*−1(*u*) is the qf of the baseline distribution and *<sup>u</sup>* ∈ (0, 1).

#### *4.5. Mean Deviations*

The amount of scattering in a population is evidently measured to some extent by the totality of deviations from the mean and median. These are known as the mean deviation about the mean and the mean deviation about the median. These measures can be calculated using the following relationships:

*δ*1(*X*) = 2*μG*(*μ*) −2 *<sup>μ</sup>* <sup>−</sup><sup>∞</sup> *xg*(*x*)*dx* and *<sup>δ</sup>*2(*X*) = *<sup>μ</sup>* <sup>−</sup><sup>2</sup> *M* <sup>−</sup><sup>∞</sup> *xg*(*x*)*dx*, respectively, where *μ* = *E*(*X*) and *M* = *Q* 1 2 .
