On a New Stochastic Ordering and Aging Classes Based on the Generalized Moment-Generating Function: Theory and Applications

SYNOPTIC ABSTRACT

This article opens up new aging classes and new stochastic orders dependant on generalized moment-generating function, which plays a vital role in reliability theory, finance topics, stochastic orders, and economic theory. The article presents some important new implications concerning these classes and introduces characterizations of the Weibull distribution, exponential distribution, and Erlang distribution through newly suggested aging classes. In addition, we list a series of inequalities that provide bounds for generalized moment-generating classes. Furthermore, a sufficient condition for allowing a probability distribution to have a new class is provided. The article also demonstrates the preservation properties of a new stochastic order under certain reliability operations, such as mixture and convolution. Moreover, some novel applications of classes and stochastic orders in random sum and shock models are provided.

KEY WORDS AND PHRASES:

• Stochastic orders
• moment-generating function
• generalized moment-generating function
• residual lifetime
• reversed residual lifetime

1. Introduction and Motivation

Consider the probability density function $f_1(.)$ for a lifetime random variable X₁ with the distribution function $F_1(.),$ survival function ${\overline{F}}_1(.)=1-F_1(.),$ and hazard rate function $r_{F_1}(.)=f_1(.)/{\overline{F}}_1(.).$ In addition, we assume that the mean life ${\mu }_{X_1}={\int }_0^{\infty }{\overline{F}}_1\left(u\right)du$ and the variance ${\sigma }_{X_1}^2$ are finite. Likewise, let X₂ be the second lifetime random variable with the density function $f_2(.)$, distribution function $F_2(.)$, survival function ${\overline{F}}_2(.)=1-F_2(.)$, and hazard rate function $r_{F_2}(.)=f_2(.)/{\overline{F}}_2(.)$. Furthermore, both the mean life ${\mu }_{X_2}={\int }_0^{\infty }{\overline{F}}_2\left(u\right)du$ and variance ${\sigma }_{X_2}^2$ are assumed to be finite. Let $l_{X_1}=\inf \left\{x_1\in {\mathbb{R}}^{+}:F_1\left(x_1\right)>0\right\},$ $u_{X_1}=\sup \left\{x_1\in {\mathbb{R}}^{+}:F_1\left(x_1\right)<1\right\},$ ${\Omega }_{X_1}=\left(l_{X_1},u_{X_1}\right),$ $l_{X_2}=\inf \left\{x_2\in {\mathbb{R}}^{+}:F_2\left(x_2\right)>0\right\},$ $u_{X_2}=\sup \left\{x_2\in {\mathbb{R}}^{+}:F_2\left(x_2\right)<1\right\}$ and ${\Omega }_{X_2}=\left(l_{X_2},u_{X_2}\right)$. Let $X_1\left(X_2\right)$ has a reversed hazard rate function $r_{F_1}(.)=f_1(.)/F_1(.)$ (${\tilde{r}}_{F_2}(.)=f_2(.)/F_2(.)$). The moment-generating functions for all $\xi \in {\mathbb{R}}^{+}$ are defined as $$M_{F_1}(\xi )={\int }_{F_1}{e}^{\xi u}d{F}_1\left(u\right)\text{and}{M}_{F_2}(\xi )={\int }_{F_2}{e}^{\xi u}d{F}_2\left(u\right).$$

Indeed, moment-generating class and stochastic ordering have received considerable attention in the literature; see ArdiÇ, Akdemir, and Özdemir (2018), Zamani, Mohtashami Borzadaran, and Amini (2017), Kayid, Izadkhah, and Alshami (2016), Rajan and Tepedelenlioğlu (2015), Gandotra, Gupta, and Bajaj (2014), Mahmoud, Moshref, and Gadallah (2014), Lee and Tepedelenlioglu (2014), Rajan (2014), Anis (2011), Elbatal (2007), Ahmed and Kayid (2004), Belzunce, Ortega, and Ruiz (2004), and Klar and Müller (2003).

Recently, Michael, Anene, Akudo, and Nwabueze (2017) defined the generalized moment-generating function as, $${\mathbb{M}}_{f_1}(\xi )={\int }_{F_1}{e}^{\xi \left(u^c+\pi \right)}d{F}_1\left(u\right)\text{for}\text{all}\xi ,c,\pi \in \mathbb{R}.$$

Now, we can define the generalized Laplace transform function as, (1) $${\mathbb{M}}_{f_1}(\xi )={\int }_{F_1}{e}^{-\xi \left(u^c+\pi \right)}d{F}_1\left(u\right),{\mathbb{M}}_{f_2}(\xi )={\int }_{F_2}{e}^{-\xi \left(u^c+\pi \right)}d{F}_2\left(u\right),$$(1) (2) $${\mathbb{M}}_{f_1}^{\ast }(\xi )={\int }_{F_1}{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du\text{and}{\mathbb{M}}_{f_2}^{\ast }(\xi )={\int }_{F_2}{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_2\left(u\right)du.$$(2)

Let us observe that, (3) $${\mathbb{M}}_{f_1}(\xi )=1+\xi c{\int }_{F_1}{u}^{c-1}{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du.$$(3)

For any life variable $X\ge 0$, the residual life variable $X_{\alpha }=\left[X-\alpha |X\ge \alpha \right]$, where $\alpha \in \left(0,l_X\right)$ and $l_{\alpha }=\sup \left\{\alpha \ni F_X(\alpha )<1\right\}$ is a nonnegative random variable representing the remaining life of X at age α. Hence, if $F(.)$ is the distribution function of X and $\overline{F}(.)=1-F(.)$ is its survival function, the survival function of $X_{\alpha }$ is thus given by, $${\overline{F}}_{X_{\alpha }}\left(x\right)=\frac{\overline{F}\left(x+\alpha \right)}{\overline{F}(\alpha )},x\ge 0,\alpha \ge 0.$$

In realistic situations, the random variables are not necessarily related only to the future; rather, they can also refer to the past. This random variable may be referred to in the literature as inactivity time $X_{(\alpha )},$ or the time elapsed after failure until time $\alpha$, given that the unit has already failed by time α. It is also called reverse residual life, or time since failure, where $$X_{(\alpha )}=[\alpha -X|X\le \alpha ],$$ for fixed $\alpha >0$, having the distribution function $$F_{(\alpha )}\left(s\right)=P[\alpha -X\le s|X\le \alpha ]=\frac{F(\alpha )-F\left(\alpha -s\right)}{F(\alpha )},$$

The reliability function here can be represented as, $${\overline{F}}_{(\alpha )}\left(s\right)=P[\alpha -X>s|X\le \alpha ]=\frac{F\left(\alpha -s\right)}{F(\alpha )}.$$

Let $X_1(\alpha )$ and $X_2(\alpha )$ be residual lives at age α of the random variables X₁ and X₂, respectively, and let ${\tilde{X}}_1(\alpha )$ and ${\tilde{X}}_2(\alpha )$ be reversed residual lives at age $\alpha$ of the random variables X₁ and X₂, respectively.

Definition 1.

(Shaked and Shanthikumar, 2007, Kayid and Alamoudi, 2013, and Ali, 2018). X₁ is smaller than or equal to X₂ in the following cases.

1. The moment-generating function ordering ($X_1{\le }_{mg}{X}_2$), if $M_{F_1}(\xi )\le M_{F_2}(\xi ),$ for all $\xi \in {\mathbb{R}}^{+},$ or ${\int }_{F_1}{e}^{-\xi u}{\overline{F}}_1\left(u\right)du$ and $M_{F_2}(\xi )={\int }_{F_2}{e}^{-\xi u}{\overline{F}}_2\left(u\right)du$ for all $\xi \in {\mathbb{R}}^{+}.$

2. The moment-generating function ordering of residual lives ($X_1(\alpha ){\le }_{mg-rl}{X}_2(\alpha )$), if ${\int }_{\alpha }^{\infty }{e}^{\xi u}{\overline{F}}_1\left(u\right)du/e^{\alpha \xi }{\overline{F}}_1(\alpha )\le {\int }_{\alpha }^{\infty }{e}^{\xi u}{\overline{F}}_2\left(u\right)du/e^{\alpha \xi }{\overline{F}}_2(\alpha )$ for all $\alpha ;\xi \in {\mathbb{R}}^{+}.$

3. The Laplace transform function ordering of residual lives ($X_1(\alpha ){\le }_{lr-rl}{X}_2(\alpha )$), if ${\int }_{\alpha }^{\infty }{e}^{-\xi u}{\overline{F}}_1\left(u\right)du/e^{-\alpha \xi }{\overline{F}}_1(\alpha )\le {\int }_{\alpha }^{\infty }{e}^{-\xi u}{\overline{F}}_2\left(u\right)du/e^{-\alpha \xi }{\overline{F}}_2(\alpha )$ for all $\alpha ;\xi \in {\mathbb{R}}^{+}.$

4. The increasing convex (concave) ordering ($X_1{\le }_{icx}\left({\le }_{icv}\right)X_2$), if $E\left[\varphi \left(X_1\right)\right]\le E\left[\varphi \left(X_2\right)\right],$ for all increasing convex (concave) functions $\varphi :\mathfrak{R}\to \mathfrak{R},$ and provided the expectations existed.

5. The exponential of inactivity time ordering ($X_1(\alpha ){\le }_{\hspace{0.17em}\text{exp}\hspace{0.17em}-it}{X}_2(\alpha )$), if ${\int }_0^{\alpha }{e}^{-\xi u}{F}_1\left(u\right)du/e^{-\alpha \xi }{F}_1(\alpha )\le {\int }_0^{\alpha }{e}^{-\xi u}{F}_2\left(u\right)du/e^{-\alpha \xi }{F}_2(\alpha )$ for all $\alpha ;\xi \in {\mathbb{R}}^{+}.$

6. The hazard rate ordering ($X_1{\le }_{hr}{X}_2$), if $r_{F_1}(.)\le r_{F_2}(.),$ for all $\xi \in {\mathbb{R}}^{+},$ or ${\overline{F}}_1(\xi )/{\overline{F}}_2(\xi )$ for all $\xi \in {\mathbb{R}}^{+}.$

7. The reversed hazard rate ordering ($X_1{\le }_{rh}{X}_2$), if ${\tilde{r}}_{F_1}(.)\le {\tilde{r}}_{F_2}(.),$ for all $\xi \in {\mathbb{R}}^{+},$ or $F_1(\xi )/F_2(\xi )$ for all $\xi \in {\mathbb{R}}^{+}.$

Now, we can define the generalized moment-generating (GL) function order as follows.

Definition 2.

Let X₁ and X₂ be two non-negative and absolutely continuous random variables, with the distribution functions $F_1(.)$ and $F_2(.)$, and the density functions $f_1(.)$ and $f_2(.)$, respectively. Hence, X₁ is smaller than or equal to X₂ in generalized moment-generating ordering, denoted by $X_1{\le }_{\tilde{G}}{X}_2$ if, (4) $${\mathbb{M}}_{f_1}(\xi )\le {\mathbb{M}}_{f_2}(\xi ),$$(4) for all $\alpha \in \left(0,u_{X_1}\right)\cap \left(0,u_{X_2}\right).$

In addition, we can define the generalized moment-generating function class of residual (reversed residual) lives as follows.

Definition 3.

Let X₁ be a non-negative and absolutely continuous random variables, with the distribution function $F_1(.)$, the survival function ${\overline{F}}_1(.)$, and the density function f₁ $(.)$. Hence,

• The generalized moment-generating function of residual lives, for all $\xi \in {\mathbb{R}}^{+}$ is defined as, (5) $${\mathbb{M}}_{X_t}(\xi )={\int }_t^{\infty }\frac{e^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)}{e^{-\xi \left(t^c+\pi \right)}{\overline{F}}_1\left(t\right)}du.$$(5)

• The generalized moment-generating function of reversed residual lives, for all $\xi \in {\mathbb{R}}^{+},$ is defined as, (6) $${\mathbb{M}}_{X_{\left(t\right)}}(\xi )={\int }_0^t\frac{e^{-\xi \left(u^c+\pi \right)}{F}_1\left(u\right)}{e^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)}du.$$(6)

Next, we consider new aging classes following previous procedures for the generalized moment-generating function class of residual (reversed residual) lives.

Definition 4.

X is said to have,

• Increasing (decreasing) the generalized moment generation of residual lives class ($X\in IG{r}_{\uparrow }$ $\left(X\in DG{r}_{\uparrow }\right))$ if, (7) $${\mathbb{M}}_{X_{t_1}}(\xi )\le (\ge ){\mathbb{M}}_{X_{t_2}}(\xi ),\text{for}\text{all}{t}_1\le t_2.$$(7)

• Increasing (decreasing) the generalized moment generation of reversed residual lives class ($X\in IG{r}_{\downarrow }$ $\left(X\in DG{r}_{\downarrow }\right))$ if, (8) $${\mathbb{M}}_{X_{\left(t_1\right)}}(\xi )\ge (\le ){\mathbb{M}}_{X_{\left(t_2\right)}}(\xi ),\text{for}\text{all}{t}_1\le t_2.$$(8)

Definition 5.

Let U and V be two non-negative and absolutely continuous random variables, with the distribution functions $\mathbb{M}(.)$ and $\mathbb{K}(.)$, density functions f $(.)$ and $g(.)$, generalized moment generation of residual lifetime functions ${\mathbb{M}}_{U_t}(\xi )$ and ${\mathbb{M}}_{V_t}(\xi )$, for t > 0, and generalized moment generation of reversed residual lifetime functions ${\mathbb{M}}_{U_{\left(t\right)}}(\xi )$ and ${\mathbb{M}}_{V\left(t\right)}(\xi )$, for t > 0, respectively. Hence,

• U is smaller than or equal to V in the generalized moment generation of residual lives ordering, denoted by $U{\le }_{\tilde{G}-rl\uparrow }V$ if, (9) $${\mathbb{M}}_{U_t}(\xi )\le {\mathbb{M}}_V(\xi ),\text{for}\text{all}t>0.$$(9)

• U is smaller than or equal to V in the generalized moment generation of reversed residual lives ordering, denoted by $U{\le }_{\tilde{G}-rl\downarrow }V$ is defined as, for all $\xi \in {\mathbb{R}}^{+}:$ (10) $${\mathbb{M}}_{U_{\left(t\right)}}(\xi )\le {\mathbb{M}}_{V\left(t\right)}(\xi ),\text{for}\text{all}t>0.$$(10)

Definition 6.

A function f is said to be star-shaped (ani-star-shaped) if $f(0)=0$ and $f(x)/x$ is increasing (decreasing) in $x\ge 0.$

For an application of the new class GL in the field of insurance, if we represent the distribution by the appropriate random variable Y and let β present the risk measure functional, therefore, $$\beta :\mathbf{Y}\to \mathbb{R}.$$

Let an insurance contract in a specified period $\left(0,a\right)$ and $\Omega$ be the state space. If none of the risks specified in the policy contract occur during the policy term, then the policy holder receives no monetary compensation for the paid premiums. The loss in compensation that occurs is the difference between the amount of compensation $\left(\text{that}\text{denoted}\text{by}\mathbb{D}\right)$ and the total losses resulting from achieve state θ (denoted by $\psi \left({\mathbf{Y}}_{(\theta )}\right)=e^{-\xi \left({\mathbf{Y}}^c+\pi \right)}$ is log-concave function) is $[\mathbb{D}-\psi \left({\mathbf{Y}}_{(\theta )}\right)|{\mathbf{Y}}_{(\theta )}\le \mathbb{D}]$, where Y is the premium state θ. Suppose that we associate a risk as $E[\mathbb{D}-\psi \left({\mathbf{Y}}_{(\theta )}\right)|{\mathbf{Y}}_{(\theta )}\le \mathbb{D}]$, which is convex. We define it using definition 3; thus, it is natural to demand that the risk function includes this as non-decreasing. Then, the expected random variable $[\mathbb{D}-\psi \left({\mathbf{Y}}_{(\theta )}\right)|{\mathbf{Y}}_{(\theta )}\le \mathbb{D}]$ based on the convex function, $\psi \left({\mathbf{Y}}_{(\theta )}\right),$ can be used to study the effects of both investor and analyst beliefs on securities trading. Moreover, we can use GL to measure the dispersion of returns for an investment portfolio. In addition, GL measures the variability of a security’s returns relative to the market index or a particular benchmark. If $E[\mathbb{D}-\psi \left({\mathbf{Y}}_{(\theta )}\right)|{\mathbf{Y}}_{(\theta )}\le \mathbb{D}]$ is greater than that of the benchmark, then the financial instrument is thought to be more perilous than the benchmark price. Low $E[\mathbb{D}-\psi \left({\mathbf{Y}}_{(\theta )}\right)|{\mathbf{Y}}_{(\theta )}\le \mathbb{D}]$ may indicate lower risk.

There is also the possibility of applying the new class in the field of investment; for example, admissible investment strategies, which can be defined by $E\left[\psi \left({\mathbf{Y}}_{(\theta )}\right)\right]=E\left[e^{-\xi \left({\mathbf{Y}}^c+\pi \right)}\right]$, and is defined in Equations (2)(2) $${\mathbb{M}}_{f_1}^{\ast }(\xi )={\int }_{F_1}{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du\text{and}{\mathbb{M}}_{f_2}^{\ast }(\xi )={\int }_{F_2}{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_2\left(u\right)du.$$(2) and (3)(3) $${\mathbb{M}}_{f_1}(\xi )=1+\xi c{\int }_{F_1}{u}^{c-1}{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du.$$(3) .

As a result, this study has three aims. The first aim is to consider new nonparametric classes of distributions ($IG{r}_{\uparrow },DG{r}_{\uparrow },IG{r}_{\downarrow }$, and $DG{r}_{\downarrow }$), those dependent on ${\mathbb{M}}_{f_1}(\xi ),{\mathbb{M}}_{X_t}(\xi )$, and ${\mathbb{M}}_{X_{\left(t\right)}}(\xi )$, while introducing certain characterizations and preservations. The second aim is to suggest a new technique that improves comparison between two distributions with certain characterizations and thus, preservations, which depend upon ${\le }_{\tilde{G}},$ ${\le }_{\tilde{G}-rl\uparrow }$ and ${\le }_{\tilde{G}-rl\downarrow }.$ The third goal is to apply these results in random sum and shock models. Section 2 provides sufficient conditions for new classes, introduces their useful properties and characterizations, and studies certain characterizations of the Weibull, exponential, and Erlang distributions through newly suggested aging classes. In addition, we offer a series of inequalities that provide the bounds for $M_{f_1}(\xi ),$ $M_{X_t}(\xi )$ and $M_{X_{\left(t\right)}}(\xi )$ functions. In Section 3, relationships between the strong risk order and other stochastic orders are discussed. Furthermore, useful properties and characterizations of the ${\le }_{\tilde{G}},$ ${\le }_{\tilde{G}-rl\uparrow }$ and ${\le }_{\tilde{G}-rl\downarrow }$ orders are analyzed. We also establish closure properties of ${\le }_{\tilde{G}},$ ${\le }_{\tilde{G}-rl\uparrow }$ and ${\le }_{\tilde{G}-rl\downarrow }$ orders under relevant reliability operations, such as convolution and mixture. Finally, Section 4 discusses useful applications in random sum and shock models by using new aging classes involving the ${\le }_{\tilde{G}},$ ${\le }_{\tilde{G}-rl\uparrow }$ and ${\le }_{\tilde{G}-rl\downarrow }$ orders.

2. Characterizations of GL Class

By using G for generalized moment-generating residual life, $\mathbf{D}$ for decreasing, and I for increasing, we can define the following two classes of a lifetime.

1. ${\mathbf{G}}_{{\mathbf{D}}_{\uparrow }}=\left(F\in DG{r}_{\uparrow },\text{for}\text{all}t,\text{is}\mathrm{a}\mathbf{D}\right);$

2. ${\mathbf{G}}_{{\mathbf{I}}_{\uparrow }}=\left(F\in IG{r}_{\uparrow }\text{for}\text{all}t,\text{is}\mathrm{a}\mathbf{I}\right);$

3. ${\mathbf{G}}_{{\mathbf{D}}_{\downarrow }}=\left(F\in DG{r}_{\downarrow },\text{for}\text{all}t,\text{is}\mathrm{a}\mathbf{D}\right);$

4. ${\mathbf{G}}_{{\mathbf{I}}_{\downarrow }}=\left(F\in IG{r}_{\downarrow }\text{for}\text{all}t,\text{is}\mathrm{a}\mathbf{I}\right).$

Clearly, ${\mathbf{G}}_{{\mathbf{D}}_{\uparrow }}$ and ${\mathbf{G}}_{{\mathbf{I}}_{\uparrow }}$ form a pair of dual classes based on distinguishing ${\mathbb{M}}_{X_t}(\xi )$ and ${\mathbf{G}}_{{\mathbf{D}}_{\downarrow }}$ and ${\mathbf{G}}_{{\mathbf{I}}_{\downarrow }}$ form a pair of dual classes based on distinguishing ${\mathbb{M}}_{X_{\left(t\right)}}(\xi )$. Next, the results reveal certain properties and applications of the ${\mathbf{G}}_{{\mathbf{D}}_{\uparrow }}$ and ${\mathbf{G}}_{{\mathbf{I}}_{\uparrow }}$ classes of life distributions.

Now, we provide the sufficient conditions for a probability distribution to have the ${\mathbf{G}}_{{\mathbf{D}}_{\uparrow }}$ and ${\mathbf{G}}_{\mathbf{I}\uparrow }$ property.

Theorem 1.

Let T denote the lifetime of equipment with ${\mathbb{M}}_{X_{t_1}}(\xi )$ and ${\mathbb{M}}_{X_{t_1}}^{‵}(\xi )=\partial {\mathbb{M}}_{X_{t_1}}(\xi )/\partial t$.

1. If, (11) $${\mathbb{M}}_{X_t}(\xi )\ge (\le )1/\left\{\overline{F}\left(t\right)\left(r_F(.)+\xi c{t}^{c-1}\right)\right\}.$$(11) Then, T $\in {\mathbf{G}}_{{\mathbf{I}}_{\uparrow }}\left({\mathbf{G}}_{\mathbf{D}\uparrow }\right)$.

2. If, (12) $${\mathbb{M}}_{X_{\left(t\right)}}(\xi )\ge (\le )-1/\left[{\tilde{r}}_F\left(t\right)+\xi c{t}^{c-1}\right].$$(12)

Hence, T $\in {\mathbf{G}}_{{\mathbf{I}}_{\downarrow }}\left({\mathbf{G}}_{\mathbf{D}\downarrow }\right)$.

Proof.

It is easy to prove that, $$\begin{array}{c}{\mathbb{M}}_{X_t}^{‵}(\xi )=\frac{\partial }{\partial t}\frac{{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}\overline{F}\left(u\right)du}{e^{-\xi \left(t^c+\pi \right)}\overline{F}\left(t\right)}\\ =\frac{{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}\overline{F}\left(u\right)du\left[e^{-\xi \left(t^c+\pi \right)}f\left(t\right)+\xi c{t}^{c-1}{e}^{-\xi \left(t^c+\pi \right)}\overline{F}\left(t\right)\right]}{{\left(e^{-\xi \left(t^c+\pi \right)}\overline{F}\left(t\right)\right)}^2}-1.\end{array}$$

By using the definition of hazard rate and Equation (5)(5) $${\mathbb{M}}_{X_t}(\xi )={\int }_t^{\infty }\frac{e^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)}{e^{-\xi \left(t^c+\pi \right)}{\overline{F}}_1\left(t\right)}du.$$(5) , we can decide that, (13) $$\begin{array}{c}{\mathbb{M}}_{X_t}^{‵}(\xi )=\frac{-e^{-\xi \left(t^c+\pi \right)}+{\mathbb{M}}_{X_{t_1}}(\xi )e^{-\xi \left(t^c+\pi \right)}\left[f\left(t\right)+\xi c{t}^{c-1}\overline{F}\left(t\right)\right]}{e^{-\xi \left(t^c+\pi \right)}}\\ =\overline{F}\left(t\right){\mathbb{M}}_{X_{t_1}}(\xi )\left[r_F(.)+\xi c{t}^{c-1}\right]-1.\end{array}$$(13)

Hence, the proof is complete (i). Now, we prove the sufficient condition for a probability distribution must have the ${\mathbf{G}}_{{\mathbf{D}}_{\downarrow }}$ and ${\mathbf{G}}_{{\mathbf{I}}_{\downarrow }}$ properties. First, we have, $$\begin{array}{c}{\mathbb{M}}_{X_{\left(t\right)}}^{‵}(\xi )=\frac{\partial }{\partial t}\frac{{\int }_0^t{e}^{-\xi \left(u^c+\pi \right)}F\left(u\right)du}{e^{-\xi \left(t^c+\pi \right)}F\left(t\right)}\\ =1+\frac{{\int }_0^t{e}^{-\xi \left(u^c+\pi \right)}F\left(u\right)du\left[e^{-\xi \left(t^c+\pi \right)}f\left(t\right)+\xi c{t}^{c-1}{e}^{-\xi \left(t^c+\pi \right)}F\left(t\right)\right]}{{\left(e^{-\xi \left(t^c+\pi \right)}F\left(t\right)\right)}^2}.\end{array}$$

By using the definition of the reversed hazard rate and Equation (6)(6) $${\mathbb{M}}_{X_{\left(t\right)}}(\xi )={\int }_0^t\frac{e^{-\xi \left(u^c+\pi \right)}{F}_1\left(u\right)}{e^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)}du.$$(6) , we can decide that, (14) $$\begin{array}{c}{\mathbb{M}}_{X_{\left(t\right)}}^{‵}(\xi )=1+\frac{{\mathbb{M}}_{X_{\left(t\right)}}(\xi )\left[f\left(t\right)+\xi c{t}^{c-1}F\left(t\right)\right]}{F\left(t\right)};\\ =1+{\mathbb{M}}_{X_{\left(t\right)}}(\xi )\left[{\tilde{r}}_F\left(t\right)+\xi c{t}^{c-1}\right].\end{array}$$(14)

Therefore, the proof is complete (ii).

For X to be a non-negative and absolutely continuous random variable, with the distribution function $F(.)$ and density function f $(.)$, respectively, we denote by, $${\mathbb{M}}_f(\xi )={\int }_F{e}^{-\xi \left(u^c+\pi \right)}dF\left(u\right).$$ The generalized Laplace-Stieltjes transformation of f (GL) (or the Laplace transformation of X) and the generalized Laplace transformation of $\overline{F}$, respectively.

Next, we characterize the GL class using a function of the respective moments.

Proposition 1.

If X is a nonnegative random variable with the distribution function F and survival function $\overline{F}$, such that all its moments exist, then we get, (15) $${\mathbb{M}}_f(\xi )=e^{-\xi \pi }-\sum _{n=0}^{\infty }{\left(-1\right)}^n\frac{{(\xi )}^{n+1}}{n!}\sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right){\pi }^{n-i}\frac{E\left[X^{c(i+1)}\right]}{(i+1)}forall\xi >0.$$(15)

Proof.

By Equation (1), we have, (16) $$\begin{array}{c}{\mathbb{M}}_f(\xi )={\int }_0^{\infty }c\xi u^{c-1}{e}^{-\xi \left(u^c+\pi \right)}F\left(u\right)du\\ =e^{-\xi \pi }-{\int }_0^{\infty }c\xi u^{c-1}{e}^{-\xi \left(u^c+\pi \right)}\overline{F}\left(u\right)du,\text{for}\text{all}\xi >0.\end{array}$$(16)

By Equations (2)(2) $${\mathbb{M}}_{f_1}^{\ast }(\xi )={\int }_{F_1}{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du\text{and}{\mathbb{M}}_{f_2}^{\ast }(\xi )={\int }_{F_2}{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_2\left(u\right)du.$$(2) and (3)(3) $${\mathbb{M}}_{f_1}(\xi )=1+\xi c{\int }_{F_1}{u}^{c-1}{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du.$$(3) in Michael et al. (2017), we have, $${\mathbb{M}}_f(\xi )=e^{-\xi \pi }-\sum _{n=0}^{\infty }{\left(-1\right)}^n\frac{{(\xi )}^n}{n!}\sum _{i=0}^n\left(\begin{array}{c}n\\ i\end{array}\right){\pi }^{n-i}c\xi {\int }_0^{\infty }{u}^{c-1+ci}\overline{F}\left(u\right)du,\text{for}\text{all}\xi >0.$$

Moreover, by using Theorem 5.A.5 in Shaked and Shanthikumar (2007, pp. 235), we can complete the proof.

The Weibull distribution has been utilized in many applications, such as reliability centered maintenance, in which a shape parameter $\alpha >1\left(\alpha \le 1\right)$, allows the reliability-centered decision for preventive maintenance to be carried out (no). We use the following example to illustrate the importance of Theorem 1 in recognizing ${\mathbf{G}}_{{\mathbf{D}}_{\downarrow }}$ class.

Example 1.

Suppose that T is a Weibull random variable with the density function $f\left(t\right)=\alpha {\lambda }^{\alpha }{t}^{\alpha -1}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-{\left(\lambda t\right)}^{\alpha }\right),$ for $t>0$ and $\alpha ,\lambda >0$. One can easily prove that, (17) $${\tilde{r}}_F\left(t\right)=\left(\alpha {\lambda }^{\alpha }{t}^{\alpha -1}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-{\left(\lambda t\right)}^{\alpha }\right)\right)/\left[1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-{\left(\lambda t\right)}^{\alpha }\right)\right],$$(17) and $$e^{-\xi \left(t^c+\pi \right)}F\left(t\right)=\left[e^{-\xi \left(t^c+\pi \right)}-e^{-\xi \left(t^c+\pi \right)-{\left(\lambda t\right)}^{\alpha }}\right].$$

In addition, if $t^c>0,n=\frac{1-c}{c}\in {\mathbb{N}}^{+},$ $c=\alpha ,$ and $\gamma \left(.,.\right)$ is the lower incomplete gamma, we have, $${\int }_0^t{e}^{-\xi \left(u^c+\pi \right)}du=\frac{e^{-\xi \pi }}{c}\left[\frac{n!}{{\xi }^{n+1}}-e^{\xi t^c}\sum _{i=0}^n\frac{n!}{i!}\frac{t^{ic}}{{\xi }^{n-i+1}}\right],$$ and $${\int }_0^t{e}^{-\xi \left(u^c+\pi \right)-{\left(\lambda u\right)}^{\alpha }}du=\frac{1}{\alpha }{\left(1/\left(\xi +{\lambda }^{\alpha }\right)\right)}^{1/\alpha }{e}^{-\xi \pi }\gamma \left(\frac{1}{\alpha },\xi \left(t^{\alpha }\right)+{\left(\lambda t\right)}^{\alpha }\right).$$

By using Equation (6)(6) $${\mathbb{M}}_{X_{\left(t\right)}}(\xi )={\int }_0^t\frac{e^{-\xi \left(u^c+\pi \right)}{F}_1\left(u\right)}{e^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)}du.$$(6) , we have, $$\begin{array}{c}{\mathbb{M}}_{X_{\left(t\right)}}(\xi )\\ =\frac{1}{e^{-\xi t^{\alpha }}-e^{-\xi t^{\alpha }-{\left(\lambda t\right)}^{\alpha }}}\times \\ \left[\frac{1}{\alpha }\left[\frac{n!}{{\xi }^{n+1}}-e^{\xi t^{\alpha }}\sum _{i=0}^n\frac{n!}{i!}\frac{t^{i\alpha }}{{\xi }^{n-i+1}}\right]-\frac{1}{\alpha }{\left(1/\left(\xi +{\lambda }^{\alpha }\right)\right)}^{1/\alpha }\gamma \left(\frac{1}{\alpha },\xi \left(t^{\alpha }\right)+{\left(\lambda t\right)}^{\alpha }\right)\right],\end{array}$$ if $t^c>0,$ $n=\frac{1-c}{c}\in {\mathbb{N}}^{+},c=\alpha \in (0,1],$ and $\gamma \left(.,.\right)$ is the lower incomplete gamma. By using Equation (17)(17) $${\tilde{r}}_F\left(t\right)=\left(\alpha {\lambda }^{\alpha }{t}^{\alpha -1}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-{\left(\lambda t\right)}^{\alpha }\right)\right)/\left[1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-{\left(\lambda t\right)}^{\alpha }\right)\right],$$(17) , we can easily check that the right-hand side of inequality Equation (12)(12) $${\mathbb{M}}_{X_{\left(t\right)}}(\xi )\ge (\le )-1/\left[{\tilde{r}}_F\left(t\right)+\xi c{t}^{c-1}\right].$$(12) is $$1/\left[{\tilde{r}}_F\left(t\right)+\xi c{t}^{c-1}\right]=1/\left(\left(\left(\alpha {\lambda }^{\alpha }{t}^{\alpha -1}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-{\left(\lambda t\right)}^{\alpha }\right)\right)/\left(1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-{\left(\lambda t\right)}^{\alpha }\right)\right)\right)+\xi c{t}^{c-1}\right).$$

According to Theorem 1, this means that T $\in {\mathbf{G}}_{\mathbf{D}\downarrow }.$

The characteristics of GL will be obtained in the following results.

Theorem 2.

Let $$e^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)be\mathrm{a}\text{twice}\text{differentiable}\text{and}\hspace{0.17em}\text{log}\hspace{0.17em}-\text{concave}\text{function},$$ and (18) $$\phi \left(t\right)={\int }_0^t{e}^{-\xi \left(u^c+\pi \right)}{F}_1\left(u\right)du,t\in \left(0,\infty \right).$$(18)

Then, $\phi \left(t\right)$ is a log-concave function.

Proof.

Let, $$G\left(t\right)=\frac{{\phi }^{″}\left(t\right)\phi \left(t\right)-{\left({\phi }^{\prime }\left(t\right)\right)}^2}{\frac{\partial }{\partial t}{e}^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)},t\in \left[0,\infty \right].$$

Then, (19) $$\Phi \left(t\right)=\frac{{\left({\int }_0^t{e}^{-\xi \left(u^c+\pi \right)}{F}_1\left(u\right)du\right)}^{″}{\int }_0^t{e}^{-\xi \left(u^c+\pi \right)}{F}_1\left(u\right)du-{\left(e^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)\right)}^2}{\frac{\partial }{\partial t}{e}^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)},t\in \left[0,\infty \right].$$(19)

Therefore, Equation (19)(19) $$\Phi \left(t\right)=\frac{{\left({\int }_0^t{e}^{-\xi \left(u^c+\pi \right)}{F}_1\left(u\right)du\right)}^{″}{\int }_0^t{e}^{-\xi \left(u^c+\pi \right)}{F}_1\left(u\right)du-{\left(e^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)\right)}^2}{\frac{\partial }{\partial t}{e}^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)},t\in \left[0,\infty \right].$$(19) represents, $$\begin{array}{c}\Phi \left(t\right)=\frac{{\left(e^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)\right)}^{\prime }{\int }_0^t{e}^{-\xi \left(u^c+\pi \right)}{F}_1\left(u\right)du-{\left(e^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)\right)}^2}{\frac{\partial }{\partial t}{e}^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)},\\ ={\int }_0^t{e}^{-\xi \left(u^c+\pi \right)}{F}_1\left(u\right)du-\frac{{\left(e^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)\right)}^2}{\frac{\partial }{\partial t}{e}^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)},t\in \left[0,\infty \right].\end{array}$$

Thus, the following equation is obtained. (20) $${\Phi }^{\prime }\left(t\right)=e^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)-\frac{\partial }{\partial t}\frac{{\left(e^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)\right)}^2}{\frac{\partial }{\partial t}{e}^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)},t\in \left[0,\infty \right].$$(20)

Equation (20)(20) $${\Phi }^{\prime }\left(t\right)=e^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)-\frac{\partial }{\partial t}\frac{{\left(e^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)\right)}^2}{\frac{\partial }{\partial t}{e}^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)},t\in \left[0,\infty \right].$$(20) demonstrates that, $$\begin{array}{c}{\Phi }^{\prime }\left(t\right)=e^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)-2\left(e^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)\right)+\frac{\left(\frac{{\partial }^2}{\partial t^2}{e}^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)\right){\left(e^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)\right)}^2}{{\left(\frac{\partial }{\partial t}{e}^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)\right)}^2}\\ =-e^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)+\frac{\left(\frac{{\partial }^2}{\partial t^2}{e}^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)\right){\left(e^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)\right)}^2}{{\left(\frac{\partial }{\partial t}{e}^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)\right)}^2}\le 0.\end{array}$$

Then, $\Phi \left(t\right)$ is decreasing. In addition, we get, $$\Phi \left(t\right)\le \underset{t\to 0^{+}}{\lim }\Phi \left(t\right)=-\frac{{\left(e^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)\right)}^2}{\frac{\partial }{\partial t}{e}^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)}\le 0,$$ and $${\phi }^{″}\left(t\right)\phi \left(t\right)-{\left({\phi }^{\prime }\left(t\right)\right)}^2<0.$$

The proof is complete.

Proposition 2.

Let, $$\phi \left(t\right)={\int }_0^t{e}^{-\xi \left(u^c+\pi \right)}{F}_1\left(u\right)du.$$

Then, $$X\in DG{r}_{\downarrow }\left(IG{r}_{\downarrow }\right)\iff \phi \left(t\right)\text{is}\hspace{0.17em}\text{log}\hspace{0.17em}-\text{concave}\left(\text{convex}\right)\text{function}\text{in}t.$$

Proof.

Let, $$\begin{array}{c}X\in DG{r}_{\downarrow }\left(IG{r}_{\downarrow }\right)\iff X_{\left(t_1\right)}{\le }_{\tilde{G}}{X}_{\left(t_2\right)},\text{for}\text{all}{t}_1\le t_2.\\ \iff {\mathbb{M}}_{X_{\left(t_1\right)}}(\xi )\le (\ge ){\mathbb{M}}_{X_{\left(t_2\right)}}(\xi )\\ \iff \frac{{\int }_0^{t_1}{e}^{-\xi \left(u^c+\pi \right)}{F}_1\left(u\right)du}{e^{-\xi \left(t_1^c+\pi \right)}{F}_1\left(t_1\right)}\le (\ge )\frac{{\int }_0^{t_2}{e}^{-\xi \left(u^c+\pi \right)}{F}_1\left(u\right)du}{e^{-\xi \left(t_2^c+\pi \right)}{F}_1\left(t_2\right)}\\ \iff -\frac{e^{-\xi \left(t_2^c+\pi \right)}{F}_1\left(t_2\right)}{{\int }_0^{t_2}{e}^{-\xi \left(u^c+\pi \right)}{F}_1\left(u\right)du}\ge (\le )-\frac{e^{-\xi \left(t_1^c+\pi \right)}{F}_1\left(t_1\right)}{{\int }_0^{t_1}{e}^{-\xi \left(u^c+\pi \right)}{F}_1\left(u\right)du}.\end{array}$$

The proof is complete.

Exponential distribution is extensively and widely used in reliability engineering to model the lifetime of electronic and electrical components and systems.

We use the following example to illustrate the importance of Theorem 2 in recognizing ${\mathbf{G}}_{{\mathbf{D}}_{\downarrow }}$ class.

Example 2.

Suppose that T is an exponential random variable with the density function $f\left(t\right)=\lambda \hspace{0.17em}\text{exp}\hspace{0.17em}-\lambda t,$ and distribution function, (21) $$F\left(t\right)=1-\hspace{0.17em}\text{exp}\hspace{0.17em}-\lambda t,$$(21) for t > 0, and $\lambda >0$. Let, (22) $$\varphi \left(\xi ,\pi ,t\right)=e^{-\xi \left(t+\pi \right)}F\left(t\right).$$(22)

Substituting Equation (21)(21) $$F\left(t\right)=1-\hspace{0.17em}\text{exp}\hspace{0.17em}-\lambda t,$$(21) in to Equation (22)(22) $$\varphi \left(\xi ,\pi ,t\right)=e^{-\xi \left(t+\pi \right)}F\left(t\right).$$(22) , we obtain $$\varphi \left(\xi ,\pi ,t\right)=e^{-\xi \left(t^c+\pi \right)}(1-e^{-\lambda t}),$$

The second derivative of $\varphi \left(\xi ,\pi ,t\right)$ is obtained as, (23) $${\varphi }^{‵‵}\left(\xi ,\pi ,t\right)={\xi }^2{e}^{-\xi \left(t+\pi \right)}(1-e^{-\lambda t})-2\xi \lambda e^{-\xi \left(t+\pi \right)-\lambda t}-{\lambda }^2{e}^{-\xi \left(t+\pi \right)-\lambda t}.$$(23)

In addition, we have, (24) $$\varphi \left(\xi ,\pi ,\left(\theta t+\left(1-\theta \right)u\right)\right)\ge {\left(\varphi \left(\xi ,\pi ,t\right)\right)}^{\theta }{\left(\varphi \left(\xi ,\pi ,u\right)\right)}^{1-\theta },$$(24) for $t,u\in {\mathbb{R}}^{+}$ and $\theta \in \left[0,1\right].$ From relation (24), we deduce that $\varphi \left(\xi ,\pi ,t\right)$ is twice differentiable with a log-concave function. It remains to be proven whether ${\int }_0^t\varphi \left(\xi ,\pi ,u\right)du$ is a log concave or logarithmically concave function. When we know that, $$\begin{array}{c}l\left(\xi ,\pi ,t\right)={\int }_0^t\varphi \left(\xi ,\pi ,u\right)du\\ ={\xi }^{-1}{e}^{-\xi \left(t+\alpha \right)}\left(e^{t\xi }-1\right)-{\left(\xi +\lambda \right)}^{-1}{e}^{-\alpha \xi -t\left(\xi +\lambda \right)}\left(e^{t\left(\xi +\lambda \right)}-1\right).\end{array}$$

We can now proceed analogously to $\varphi \left(\xi ,\pi ,t\right)$, and we can deduce that $l\left(\xi ,\pi ,t\right)$ is logarithmically concave. From Proposition 2, we see that $T\in DG{r}_{\downarrow }$.

Proposition 3.

Let, $$\overline{\phi }\left(t;\xi ,c,\pi \right)={\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du.$$

Then, $$X\in DG{r}_{\uparrow }\left(IG{r}_{\uparrow }\right)\iff \overline{\phi }\left(t\right)\text{is}\hspace{0.17em}\text{log}\hspace{0.17em}-\text{concave}\left(\text{convex}\right)\text{function}\text{in}t.$$

Proof.

Let, $$\begin{array}{c}X\in DG{r}_{\uparrow }\left(IG{r}_{\uparrow }\right)\iff X_{t_1}{\le }_{\tilde{G}}{X}_{t_2},\text{for}\text{all}{t}_1\le t_2.\\ \iff {\mathbb{M}}_{X_{t_1}}(\xi )\le (\ge ){\mathbb{M}}_{X_{t_2}}(\xi )\\ \iff \frac{{\int }_{t_1}^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du}{e^{-\xi \left(t_1^c+\pi \right)}{\overline{F}}_1\left(t_1\right)}\le (\ge )\frac{{\int }_{t_2}^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du}{e^{-\xi \left(t_2^c+\pi \right)}{\overline{F}}_1\left(t_2\right)}\\ \iff -\frac{e^{-\xi \left(t_2^c+\pi \right)}{\overline{F}}_1\left(t_2\right)}{{\int }_{t_2}^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du}\ge (\le )-\frac{e^{-\xi \left(t_1^c+\pi \right)}{\overline{F}}_1\left(t_1\right)}{{\int }_{t_1}^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du}.\end{array}$$

The proof has been completed.

For several years, great effort has been devoted to the study of the Erlang distribution due to its relationship with exponential and gamma distributions and because of its uses in the field of stochastic processes. Suppose that T is an Erlang random variable with the density function, $$f\left(t;b,a\right)=\frac{a^b{t}^{b-1}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-at\right)}{\left(b-1\right)!},\text{for}t,a\ge 0.$$ where $“\hspace{0.17em}\text{exp}\hspace{0.17em}”$ is the base of the natural logarithm and $!$ is the factorial function. Parameter b is called the shape parameter, and parameter a is called the rate (scale) parameter; the cumulative distribution function of the Erlang distribution can be expressed as, $$F\left(t;b,a\right)=1-\sum _{i=0}^{b-1}\frac{{\left(at\right)}^i}{i!}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-at\right).$$

Now, we can discuss the behavior of $\overline{\phi }\left(t\right)$ in the Erlang distribution.

Example 3.

Suppose that T belongs to the Erlang family with the shape parameter b and rate (scale) parameter a,; then, we have, $$\begin{array}{c}\overline{\phi }\left(t;\xi ,1,\pi \right)={\int }_t^{\infty }{e}^{-\xi \left(u+\pi \right)}(1-F\left(u;b,a\right))du,\\ =e^{-\xi \pi }\sum _{i=0}^{b-1}\frac{a^i}{i!}{\int }_t^{\infty }{u}^i{e}^{-\xi u-au}du.\end{array}$$

According to the transformation technique, we obtain, $$\overline{\phi }\left(t;\xi ,1,\pi \right)=e^{-\xi \pi }\sum _{i=0}^{b-1}\frac{a^i}{i!}{\left(a+\xi \right)}^{-1}\left[\begin{array}{c}{\left(a+\xi \right)}^{-i}\Gamma \left(1+i\right)\\ +t^i{\left(t\left(a+\xi \right)\right)}^{-i}\left(-\Gamma \left(i+1\right)+\Gamma \left(1+i,t\left(a+\xi \right)\right)\right)\end{array}\right],$$ for $a+\xi >0.$ It easy to prove that, $$\overline{\phi }\left(\left(\theta x+\left(1-\theta \right)y\right);\xi ,1,\pi \right)\ge {\left[\overline{\phi }\left(t;\xi ,1,\pi \right)\right]}^{\theta }{\left[\overline{\phi }\left(t;\xi ,1,\pi \right)\right]}^{1-\theta },$$

for $x,y\in {\mathbb{R}}^{+}$ and $\theta \in \left[0,1\right].$ From Proposition 3, we deduce that $T\in DG{r}_{\uparrow }.$

We list a series of inequalities that provide bounds for the ${\mathbb{M}}_{f_1}(\xi ),$ ${\mathbb{M}}_{X_t}(\xi )$ and ${\mathbb{M}}_{X_{\left(t\right)}}(\xi )$ functions according to the following theorem.

Theorem 3.

If X is a nonnegative random variable with density function f and distribution function F such that all its moments exist, suppose that ${\phi }_{\chi }\left(t\right)=\chi e^{-\xi \left(t^c+\pi \right)};$ $\chi =f,F,\overline{F}$ and for $c\in \left(0,\infty \right)$ and ${\phi }_{\chi }^{\prime }\left(c\right)\ne 0;$ $\chi =f;$ $F;$ $\overline{F}.$ Then,

• ${\mathbb{M}}_{f_1}(\xi )\le \frac{-{\left({\phi }_f\left(c\right)\right)}^2}{{\phi }_f^{\prime }\left(c\right)}\text{exp}\hspace{0.17em}\left(c\frac{{\phi }_f^{\prime }\left(c\right)}{{\phi }_f\left(c\right)}\right);$

• ${\mathbb{M}}_{X_t}(\xi )\le \left[\frac{{\left({\phi }_F\left(c\right)\right)}^2}{{\phi }_F^{\prime }\left(c\right)}\left(1-\hspace{0.17em}\text{exp}\hspace{0.17em}\left(c\frac{{\phi }_F^{\prime }\left(c\right)}{{\phi }_F\left(c\right)}\right)\right)+\frac{{\left({\phi }_F\left(c\right)\right)}^2}{{\phi }_F^{\prime }\left(c\right)}\text{exp}\hspace{0.17em}\left(\left(t-c\right)\frac{{\phi }_F^{\prime }\left(c\right)}{{\phi }_F\left(c\right)}\right)\right]/e^{-\xi \left(t^c+\pi \right)}F\left(t\right);$

• ${\mathbb{M}}_{X_{\left(t\right)}}(\xi )\ge \frac{-{\left({\phi }_{\overline{F}}\left(c\right)\right)}^2}{{\phi }_{\overline{F}}^{\prime }\left(c\right)}\text{exp}\hspace{0.17em}\left(\left(c-t\right)\frac{{\phi }_{\overline{F}}^{\prime }\left(c\right)}{{\phi }_{\overline{F}}\left(c\right)}\right)/e^{-\xi \left(t^c+\pi \right)}\overline{F}\left(t\right).$

Proof.

By using Theorem (2) and Equations (1)(1) $${\mathbb{M}}_{f_1}(\xi )={\int }_{F_1}{e}^{-\xi \left(u^c+\pi \right)}d{F}_1\left(u\right),{\mathbb{M}}_{f_2}(\xi )={\int }_{F_2}{e}^{-\xi \left(u^c+\pi \right)}d{F}_2\left(u\right),$$(1) , (5)(5) $${\mathbb{M}}_{X_t}(\xi )={\int }_t^{\infty }\frac{e^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)}{e^{-\xi \left(t^c+\pi \right)}{\overline{F}}_1\left(t\right)}du.$$(5) , and (6)(6) $${\mathbb{M}}_{X_{\left(t\right)}}(\xi )={\int }_0^t\frac{e^{-\xi \left(u^c+\pi \right)}{F}_1\left(u\right)}{e^{-\xi \left(t^c+\pi \right)}{F}_1\left(t\right)}du.$$(6) , with applied Theorem 3 in Zhang and Jiang (2012), we can complete the proof.

3. Properties of GL Ordering Under Operations

Next, we characterize GL ordering by a function of respective moments.

Proposition 4.

Let X₁ and X₂ be two non-negative and absolutely continuous random variables, with the distribution functions $F_1(.)$ and $F_2(.)$, density functions $f_1(.)$ and $f_2(.),$ and survival functions ${\overline{F}}_1(.)$ and ${\overline{F}}_2(.)$, respectively. Hence, $$X_1{\le }_{\tilde{G}}{X}_2\iff \frac{{\int }_0^{\infty }{u}^{c-1}{e}^{-\xi \left(u^c+\pi \right)}{F}_1\left(u\right)du}{{\int }_0^{\infty }{u}^{c-1}{e}^{-\xi \left(u^c+\pi \right)}{F}_2\left(u\right)du}\text{is}\text{strictly}\text{decreasing}.$$

Proof.

By Equation (16)(16) $$\begin{array}{c}{\mathbb{M}}_f(\xi )={\int }_0^{\infty }c\xi u^{c-1}{e}^{-\xi \left(u^c+\pi \right)}F\left(u\right)du\\ =e^{-\xi \pi }-{\int }_0^{\infty }c\xi u^{c-1}{e}^{-\xi \left(u^c+\pi \right)}\overline{F}\left(u\right)du,\text{for}\text{all}\xi >0.\end{array}$$(16) and Lemma 2.2 in Chan, Proschan, and Sethuraman (1990), we can complete the proof.

Proposition 5.

Let X₁ and X₂ be two non-negative and absolutely continuous random variables, with the distribution functions $F_1(.)$ and $F_2(.)$, density functions $f_1(.)$ and $f_2(.),$ and survival functions ${\overline{F}}_1(.)$ and ${\overline{F}}_2(.)$, respectively. Hence, if, $$X_1{\le }_{\tilde{G}}{X}_2\iff X_1{\ge }_{icx}{X}_2.$$

Proof.

By Definition 1 (iii) and Equation (15), we can complete the proof.

Next, we check that the order ${\le }_{\tilde{G}}$ is closed under certain operations by the following results.

Proposition 6.

Let X₁ and X₂ be two non-negative and absolutely continuous random variables, with the distribution functions $F_1(.)$ and $F_2(.)$, density functions $f_1(.)$ and $f_2(.),$ and survival functions ${\overline{F}}_1(.)$ and ${\overline{F}}_2(.)$, respectively. Hence,

1. If $X_1{\le }_{\tilde{G}}{X}_2\iff X_1+d{\le }_{\tilde{G}}{X}_2+d$ for any $d\in {\mathbb{R}}^{+}.$

2. If $X_1{\le }_{\tilde{G}}{X}_2\iff d{X}_1{\le }_{\tilde{G}}d{X}_2$ for any $d\in {\mathbb{R}}^{+}$, that is, the order ${\le }_{\tilde{G}}$ is invariant under the scale transformation.

Proof.

By Equation (1)(1) $${\mathbb{M}}_{f_1}(\xi )={\int }_{F_1}{e}^{-\xi \left(u^c+\pi \right)}d{F}_1\left(u\right),{\mathbb{M}}_{f_2}(\xi )={\int }_{F_2}{e}^{-\xi \left(u^c+\pi \right)}d{F}_2\left(u\right),$$(1) and Definition 2, following the steps of Shaked and Shanthikumar (1994), 3.B.4, we can complete proofs (i) and (ii).

Theorem 4.

Let $U_1,U_2,\dots ,U_n$ and V be (n + 1) non-negative and absolutely continuous random variables. If $U_j{\le }_{\tilde{G}}V,$ therefore,

1. ${\sum }_{i=1}^{\infty }\frac{1}{i(i+1)}{U}_i{\le }_{\tilde{G}}V,$ and

2. ${\sum }_{j=1}^n{s}_j{U}_j{\le }_{\tilde{G}}V,$ if ${\sum }_{j=1}^n{s}_j=1.$

Proof.

Suppose that ψ is a convex function. Therefore, (25) $$\begin{array}{c}E\left[\psi \left(\sum _{i=1}^{\infty }\frac{1}{i(i+1)}{U}_i\right)\right]\le E\left[\sum _{i=1}^{\infty }\frac{1}{i(i+1)}\psi \left[U_i\right]\right]\\ =\sum _{i=1}^{\infty }\frac{1}{i(i+1)}E\left[\psi \left(U_i\right)\right]\\ \le \sum _{i=1}^{\infty }\frac{1}{i(i+1)}E\left[\psi \left(V\right)\right],\end{array}$$(25) since ${\sum }_{i=1}^{\infty }\frac{1}{i(i+1)}=1$ and $U_j{\le }_{\tilde{G}}V$; therefore, inequality (25) becomes, $$\sum _{i=1}^{\infty }\frac{1}{i(i+1)}E\left[\psi \left(U_i\right)\right]\le E\left[\psi \left(V\right)\right];$$

and we obtain a complete proof (a). If ${\sum }_{j=1}^n{s}_j=1$ and $$\begin{array}{c}E\left[\psi \left(\sum _{j=1}^n{s}_j{U}_j\right)\right]\le E\left[\sum _{j=1}^n{s}_j\psi \left[U_j\right]\right]=\sum _{j=1}^n{s}_{j}E\left[\psi \left(U_j\right)\right]\\ \le \sum _{j=1}^n{s}_{j}E\left[\psi \left(V\right)\right]=E\left[\psi \left(V\right)\right],\end{array}$$ then we get a complete proof (b).

Convolution is a mathematical tool with multiple applications, including statistics, computer vision, imaging, signal processing, electrical engineering, and differential equations. In probability theory, the distribution function of the sum of two independent random variables is the convolution of their individual distribution. Now, we want to clarify whether GL ordering is closed under convolution.

Theorem 5.

1. Let $U_1,U_2,\dots$ be independent, non-negative, and absolutely continuous random variables, and let $V_1,V_2,\dots$ be other independent, non-negative, and absolutely continuous random variables with $U_j{\le }_{\tilde{G}}{V}_i$. Let A and B be independent of $\left\{U_i\right\}$ and $\left\{V_i\right\}$ with A ${\le }_{\tilde{G}}B$. Hence, $$\sum _{i=1}^A{U}_i{\le }_{\tilde{G}}\sum _{i=1}^B{V}_i.$$

2. Let $U_1,U_2,\dots ,U_n$ be independent non-negative and absolutely continuous random variables, and let $V_1,V_2,\dots ,V_n$ be other independent non-negative and absolutely continuous random variables. If $U_j{\le }_{\tilde{G}}{V}_i,$ we have, $$\sum _{i=1}^n{U}_i{\le }_{\tilde{G}}\sum _{i=1}^n{V}_i,$$

that is, the order ${\le }_{\tilde{G}}$ is closed under convolution.

Proof.

This proof involved similar steps to those used in the Shaked and Shanthikumar (2007) theorem 5.C.16.

Suppose that we are given a one-parameter family of life distribution {$F_{\theta }\left(t\right),$ $t\ge 0$}, with parameter $\theta >0.$ We look at θ as a random variable and let K be its distribution function; in the literature K, it is said to be a mixing distribution. Then, the mixture H of the family $\left\{F_{\theta }\right\}$ with respect to K and the mixture $\overline{H}$ of $\left\{{\overline{F}}_{\theta }\right\}$ are defined by the following equations. $$\begin{array}{c}H\left(t\right)={\int }_0^{\infty }{\int }_0^{t}f\left(u;\theta \right)k(\theta )dud\theta ,\\ ={\int }_0^{\infty }{F}_{\theta }\left(t\right)dK(\theta ),t>0,\end{array}$$

and $$\begin{array}{c}\overline{H}\left(t\right)={\int }_0^{\infty }{\int }_t^{\infty }f\left(u;\theta \right)k(\theta )dud\theta ,\\ ={\int }_0^{\infty }{\overline{F}}_{\theta }\left(t\right)dK(\theta ),t>0.\end{array}$$

As usual, we assume that family $\left\{F_{\theta }\right\}$ and mixing distribution G satisfy certain conditions, and we want to derive the properties of the resulting mixture H.

Theorem 6.

Suppose that $F_{\theta }{\le }_{\tilde{G}}{G}_{\theta }$ for any $\theta >0$. For arbitrary mixing distribution K, then, $${\int }_{\theta }{\overline{F}}_{\theta }\left(t\right)dK(\theta ){\le }_{\tilde{G}}{\int }_{\theta }{\overline{G}}_{\theta }\left(t\right)dK(\theta ).$$

In other words, order ${\le }_{\tilde{G}}$ is in general, closed under an arbitrary mixture.

Proof.

The proof follows similar steps to those used by Shaked and Shanthikumar (1994), 3.B.4.

Next, we introduce the characteristics of ${\le }_{\tilde{G}-rl\uparrow }:$

Theorem 7.

$$U{\le }_{\tilde{G}-rl\uparrow }V\iff \frac{{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du}{{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_2\left(u\right)du}du,\text{is}\text{decreasing}\text{in}t>0\text{and}\xi ,c,\pi \in \mathbb{R}.$$

Proof.

${\mathbb{M}}_{U_t}(\xi )$ satisfies the following relation. (26) $${\mathbb{M}}_{U_t}(\xi )={\int }_t^{\infty }\frac{e^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du}{e^{-\xi \left(t^c+\pi \right)}{\overline{F}}_1\left(t\right)}=-\frac{{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du}{\frac{\partial }{\partial t}{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du},\text{for}\text{all}t>0.$$(26)

By using Equations (26)(26) $${\mathbb{M}}_{U_t}(\xi )={\int }_t^{\infty }\frac{e^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du}{e^{-\xi \left(t^c+\pi \right)}{\overline{F}}_1\left(t\right)}=-\frac{{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du}{\frac{\partial }{\partial t}{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du},\text{for}\text{all}t>0.$$(26) and (13)(13) $$\begin{array}{c}{\mathbb{M}}_{X_t}^{‵}(\xi )=\frac{-e^{-\xi \left(t^c+\pi \right)}+{\mathbb{M}}_{X_{t_1}}(\xi )e^{-\xi \left(t^c+\pi \right)}\left[f\left(t\right)+\xi c{t}^{c-1}\overline{F}\left(t\right)\right]}{e^{-\xi \left(t^c+\pi \right)}}\\ =\overline{F}\left(t\right){\mathbb{M}}_{X_{t_1}}(\xi )\left[r_F(.)+\xi c{t}^{c-1}\right]-1.\end{array}$$(13) , we have, $$\begin{array}{c}{\mathbb{M}}_{U_t}(\xi )\le {\mathbb{M}}_V(\xi )\\ \iff -\frac{{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du}{\frac{\partial }{\partial t}{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du}\le -\frac{{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_2\left(u\right)du}{\frac{\partial }{\partial t}{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_2\left(u\right)du}\\ \iff \frac{{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du}{{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_2\left(u\right)du}\text{is}\text{decreasing}\text{in}t>0\text{and}\xi ,c,\pi \in \mathbb{R}.\end{array}$$

The next result is an analog of Theorem 7; we will not detail the proof.

Theorem 8.

$$U{\le }_{\tilde{G}-rl\downarrow }V\iff \frac{{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{F}_1\left(u\right)du}{{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{F}_2\left(u\right)du},\text{is}\text{decreasing}\text{in}t>0\text{and}\xi ,c,\pi \in \mathbb{R}.$$

4. Applications of GL Ordering

Belzunce, Gao, Hu, and Pellerey (1999) examined new better (worse) than used use of the Laplace transform, NBU_Lt $\left[NW{U}_{Lt}\right]$ as follows. $$e^{-\xi t}{\overline{F}}_X\left(t\right){\int }_0^{\infty }{e}^{-\xi u}{\overline{F}}_X\left(u\right)du\ge {\int }_t^{\infty }{e}^{-\xi u}{\overline{F}}_X\left(u\right)du,$$ or $X_t{\le }_{lr-rl}X,$ for all $t\in \left(0,l_X\right)$. In addition, Belzunce et al. (1999) introduced the new Laplace transform function as, $$L^{\ast }\left(s\right)={\int }_0^{\infty }{e}^{-su}{\overline{F}}_X\left(u\right)du=\frac{1-{\int }_0^{\infty }{e}^{-su}{f}_X\left(u\right)du}{s}.$$

When c = 1 in Eq. (3)(3) $${\mathbb{M}}_{f_1}(\xi )=1+\xi c{\int }_{F_1}{u}^{c-1}{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1\left(u\right)du.$$(3) , we get, (27) $${\mathbb{M}}_{f_1}(\xi )=1+\xi e^{-\xi \pi }{\int }_{F_1}{e}^{-\xi u}{\overline{F}}_1\left(u\right)du=1+\xi e^{-\xi \pi }{L}^{\ast }(\xi ).$$(27)

Now, we can define the new better (worse) than used with respect to the generalized Laplace transform, NBU_GLt, $\left[NW{U}_{GLt}\right]$ as follows. $$e^{-\xi \left(t^c+\pi \right)}{\overline{F}}_X\left(t\right){\int }_0^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_X\left(u\right)du\ge {\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_X\left(u\right)du,$$ for all $t\in \left(0,l_X\right)$ and the exponentially better (worse) than used with respect to the generalized Laplace transformation, EBU_GLt, $\left[BW{U}_{GLt}\right]$ as follows. $$e^{-\xi \left(t^c+\pi \right)}{\overline{F}}_X\left(t\right){\int }_0^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_Y\left(u\right)du\ge {\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_X\left(u\right)du,$$ for all $t\in \left(0,l_X\right).$

The following results can be derived in an analogous manner to Proposition 2 of Belzunce et al. (1999) and Proposition 2.4 in Wang and Ma (2009); the proof is thus omitted here.

Proposition 7.

Let U be a non-negative and absolutely continuous random variable with the distribution function $F_U(.)$ and $V=h(U)$, where h is increasing. Suppose that f is the inverse of h. Hence,

1. If $U\in NB{U}_{GLt}\left[NW{U}_{GLt}\right]$ and $f(.)$ is a star-shaped (ani-star-shaped) function, then $V\in NB{U}_{GLt}\left[NW{U}_{GLt}\right].$

2. If $U\in EB{U}_{GLt}\left[EW{U}_{GLt}\right]$ and $f(.)$ is a star-shaped (ani-star-shaped) function, with $f(t)\ge (\le )t$, hence $V\in NB{U}_{GLt}\left[NW{U}_{GLt}\right].$

Now, we apply our results in random sums of variables and shock models in the following propositions.

For the following definitions, we use a compound modified geometric distribution, which possesses the discrete property of non-aging.

Definition 7.

Suppose that $\mathbb{N}\in {\mathbb{N}}^{+}$ is a random variable with the survival function ${\overline{F}}_{\mathbb{N}}\left(y\right)=Pr\left[\mathbb{N}>y\right],$ for $y\in {\mathbb{N}}^{+}$; then,

• $N\in IG{r}_{\uparrow }$ $\left(N\in DG{r}_{\uparrow }\right)$, if and if only if, $$\sum _{k=j}^{\infty }\frac{e^{-\xi \left(k^c+\pi \right)}{\phi }^{k-1}{\overline{F}}_{\mathbb{N}}\left(k\right)}{e^{-\xi \left(j^c+\pi \right)}{\phi }^{j-1}{\overline{F}}_{\mathbb{N}}\left(j\right)}\text{is}\text{increasing}\left(\text{decreasing}\right)\text{in}j:0,1,\dots ,\text{for}\text{all}\phi \in \left[0,1\right];$$

• $N\in IG{r}_{\downarrow }$ $\left(N\in DG{r}_{\downarrow }\right)$, if and if only if, $$\sum _{k=0}^j\frac{e^{-\xi \left(k^c+\pi \right)}{\phi }^{k-1}{F}_{\mathbb{N}}\left(k\right)}{e^{-\xi \left(j^c+\pi \right)}{\phi }^{j-1}{F}_{\mathbb{N}}\left(j\right)}\text{is}\text{increasing}\left(\text{decreasing}\right)\text{in}j:0,1,\dots ,\text{for}\text{all}\phi \in \left[0,1\right];$$

• $N\in NB{U}_{GLt},\left[NW{U}_{GLt}\right]$, if and if only if, $$\sum _{k=0}^{\infty }{e}^{-\xi \left(k^c+\pi \right)}{\phi }^{k-1}{\overline{F}}_{\mathbb{N}}\left(k\right)\ge (<)\sum _{k=j}^{\infty }\frac{e^{-\xi \left(k^c+\pi \right)}{\phi }^{k-1}{\overline{F}}_{\mathbb{N}}\left(k\right)}{e^{-\xi \left(j^c+\pi \right)}\left(1-\alpha \right)\left(1-\phi \right){\phi }^{j-1}{\overline{F}}_{\mathbb{N}}\left(j\right)}.$$

Definition 8.

A function f: $X\times Y\to \left[0,\infty \right)$ is said to be totally positive of order 2 $\left(T{P}_2\right)$ if for all x₁ < x₂ and y₁ < y₂ $\left(x_i\in X,y_i\in Y;i=1,2\right)$ and, $$\left|\begin{array}{cc}f\left(x_1,y_1\right)& f\left(x_1,y_2\right)\\ f\left(x_2,y_1\right)& f\left(x_2,y_2\right)\end{array}\right|\ge 0.$$

In the following results, we provide $DG{r}_{\uparrow }$ and ${\le }_{\tilde{G}-rl\uparrow }$ in random sums applications.

Theorem 9.

1. Let ${\left\{U_i\right\}}_{i=1}^{\infty }$ be independent, non-negative, and random variables, such as ${\sum }_{j=1}^N{U}_j$ $\in DG{r}_{\uparrow }$ for all $s:$ $1,2,\dots$ and let $N_1\in {\mathbb{N}}^{+}$ and N₂ $\in {\mathbb{N}}^{+},$ which are independent of U_i. If $N_1{\le }_{hr}{N}_2$, then, $$\sum _j^{N_1}{U}_j{\le }_{\tilde{G}-rl\uparrow }\sum _j^{N_2}{U}_j.$$

2. Let ${\left\{V_i\right\}}_{i=1}^{\infty }$ be independent, non-negative, and random variables, such as ${\sum }_{j=1}^N{V}_j$ $\in DG{r}_{\downarrow }$ for all $s:$ $1,2,\dots ,$ and let $N_1\in {\mathbb{N}}^{+}$ and N₂ $\in {\mathbb{N}}^{+},$ which are independent of U_i. If $N_1{\le }_{rh}{N}_2$, then, $$\sum _j^{N_1}{V}_j{\le }_{\tilde{G}-rl\downarrow }\sum _j^{N_2}{V}_j.$$

Proof.

Random variable $X_j={\sum }_{j=1}^{N_j}{U}_j$ belongs to the compound modified geometric survival function $${\overline{F}}_{\mathbb{X}}\left(y\right)=Pr\left[\mathbb{X}>y\right]=\sum _{j=1}^{\infty }\left(1-\alpha \right)\left(1-\phi \right){\phi }^{j-1}{\overline{F}}^{\ast n}\left(y\right),$$ for $y\in R^{+},0<\alpha <1,$ $0<\phi <1,Pr\left[N_j=n\right]$=$\left(1-\alpha \right)\left(1-\phi \right){\phi }^{n-1}$, and ${\overline{F}}^{\ast n}\left(y\right)=\mathrm{Pr}\left(U_1+U_2+\dots +U_n>y\right)$, where $\left\{U_1+U_2+\dots +U_n\right\}$ is an independent and identically distributed sequence of positive random variable. Since ${\sum }_j^{N_j}{U}_j$ $\in DG{r}_{\uparrow }$, we also have, $${\int }_j^{\infty }{e}^{-\xi \left(k^c+\pi \right)}{\overline{F}}^{\ast n}\left(k\right)dkisT{P}_{2}inj:0,1,\dots ,\text{for}\text{all}\phi \in \left[0,1\right].$$

In addition, by Lemma 2.1(b) in Pellerey (1993), we obtain ${\sum }_{j=1}^{\infty }\left(1-\alpha \right)\left(1-\phi \right){\phi }^{j-1}$ is TP₂. By Proposition 4 and Definition 1(vi), we have, $$\frac{{\sum }_{j=1}^{\infty }\left(1-\alpha \right)\left(1-\phi \right){\phi }^{j-1}{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_1^{\ast j}\left(u\right)du}{{\sum }_{j=1}^{\infty }\left(1-\alpha \right)\left(1-\phi \right){\phi }^{j-1}{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{F}}_2^{\ast j}\left(u\right)du}\text{is}\text{decreasing}\text{in}t.$$

We then have the complete proof of (i), and the proof of (ii) can be obtained using a similar argument.

Next, we introduce some new applications of classes and stochastic orders in the sums of independent exponential random variables via Wilks’ integral representation.

Theorem 10.

Let $\left\{U_1,U_2,\dots ,U_N\right\}$ be independent and identically exponential random variables with parameters $\left({\lambda }_i,i=1,2,\dots ,N\right);$ suppose that $N\in {\mathbb{N}}^{+}$ is a random variable with a modified geometric density function that is independent of ${\left\{U_i\right\}}_{i=1}^{\infty }.$ Let, (28) $${\rm Y}\left(t\right)=\frac{\sum _{k=0}^{\infty }{\left(-1\right)}^k\frac{\sqrt{\xi }}{k!}\left[A_{\left(t\right)}-B_{\left(t\right)}\right]}{e^{-\xi t^c}{\overline{G}}^{\ast k}\left(t\right)},$$(28) where ${\overline{G}}^{\ast k}\left(y\right)=\mathrm{Pr}\left({\sum }_{j=1}^k{U}_j\le y<{\sum }_{j=1}^{k+1}{U}_j\right),{\alpha }_{\left(s\right)}:=\gamma \left(s\right)/\left(\vartheta \left(s\right)-\gamma {\left(s\right)}^2\right),$ ${\beta }_{\left(s\right)}:=\left(1-\gamma \left(s\right)\right)\left(\gamma \left(s\right)-\vartheta \left(s\right)\right)/\left(\vartheta \left(s\right)-\gamma {\left(s\right)}^2\right)$, $\gamma \left(s\right):={\prod }_{i=1}^s{\lambda }_i\left({\lambda }_i+1\right),\vartheta \left(s\right):={\prod }_{i=1}^s{\lambda }_i\left({\lambda }_i+1\right)/\left({\lambda }_i+2\right)$, (29) $$A_{\left(t\right)}=\frac{1}{\beta }\sum _{i=0}^{\infty }\frac{{\beta }_k\Gamma \left({\alpha }_k+{\beta }_k+i\right)}{i!\left({\beta }_k+i\right)}\sum _{l=0}^b\sum _{r=0}^i\left(\begin{array}{c}b\\ l\end{array}\right)\left(\begin{array}{c}i\\ r\end{array}\right){\left(-1\right)}^{r+b-l}\pi \left(t\right),$$(29) (30) $$\begin{array}{c}{B}_{\left(t\right)}=\frac{1}{\beta {\beta }_{k+1}}\sum _{j=0}^{2b}\left(\begin{array}{c}2b\\ j\end{array}\right){\left(-1\right)}^{2b-j}\sum _{b=0}^{\infty }\frac{\Gamma \left({\alpha }_k+{\beta }_k+b\right)\Gamma \left({\beta }_k+b\right){\beta }_k}{b!\Gamma \left({\beta }_k+1+b\right)}\\ \times \sum _{a=0}^{\infty }\frac{\Gamma \left({\beta }_k+1\right)\Gamma \left({\alpha }_{k+1}+{\beta }_{k+1}+a\right)\Gamma \left({\beta }_{k+1}+a\right)}{a!\Gamma \left({\beta }_k+1+a\right)\Gamma \left({\beta }_{k+1}\right)}\\ \times \sum _{l=0}^b\sum _{i=0}^a\left(\begin{array}{c}b\\ l\end{array}\right)\left(\begin{array}{c}a\\ i\end{array}\right){\left(-1\right)}^{l+i}\overline{\pi }\left(t\right),\\ \pi \left(t\right)={\int }_t^{\infty }{u}^{c/2}{e}^{u(l+r)}du,\\ =-\frac{1}{r+1}\left[\begin{array}{c}{\left(-r-1\right)}^{-c/2}\Gamma \left(\frac{c+2}{2}\right)+t^{c/2}{\left(-t\left(1+r\right)\right)}^{-c/2}\\ \times \left(\Gamma \left(\frac{c+2}{2},-\left(r+1\right)t\right)-\Gamma \left(\frac{c+2}{2}\right)\right)\end{array}\right];r<-1,\end{array}$$(30) and $$\begin{array}{c}\overline{\pi }\left(t\right)={\int }_t^{\infty }{u}^{c/2}{e}^{u(i+j+l)}du\\ =-\frac{1}{i+j+1}\left[\begin{array}{c}{\left(-i-j-1\right)}^{-c/2}\Gamma \left(\frac{c+2}{2}\right)+t^{c/2}{\left(-t\left(1+i+j\right)\right)}^{-c/2}\\ \times \left(\Gamma \left(\frac{c+2}{2},-\left(i+j+1\right)t\right)-\Gamma \left(\frac{c+2}{2}\right)\right)\end{array}\right];i+j<-1.\end{array}$$

Then,

1. If ${\rm Y}\left(t\right)$ is decreasing (increasing) in t, then $X={\sum }_{i=1}^N{U}_i\in DG{r}_{\uparrow }$ $\left(N\in IG{r}_{\uparrow }\right);$

2. If ${\rm Y}\left(0\right)\ge (\le ){\rm Y}\left(t\right)$ is decreasing (increasing) for all t > 0, then $X={\sum }_{i=1}^N{U}_i\in NB{U}_{GLt}\left[NW{U}_{GLt}\right].$

Proof.

The random variable $X={\sum }_{i=1}^N{U}_i$ belongs to the compound modified geometric survival function, (31) $$\begin{array}{c}{\overline{K}}_{\mathbb{X}}\left(y\right)=\sum _{j=1}^{\infty }\left(1-\alpha \right)\left(1-\phi \right){\phi }^{j-1}{\overline{G}}^{\ast k}\left(y\right),\\ =\sum _{j=1}^{\infty }\Psi \left(j\right){\overline{G}}^{\ast k}\left(y\right),\end{array}$$(31) for $y\in R^{+},$ $0<\alpha <1,$ $0<\phi <1$, $$Pr\left[N=s\right]=\Psi \left(s\right)=\left(1-\alpha \right)\left(1-\phi \right){\phi }^{s-1},\alpha ;\phi ;s>0,$$ and $${\overline{G}}^{\ast k}\left(y\right)=\mathrm{Pr}\left(\sum _{j=1}^k{U}_j\le y<\sum _{j=1}^{k+1}{U}_j\right),$$ where $\left\{U_1,U_2,\dots ,U_N\right\}$ are independent and identically exponential random variables with parameters $\left({\lambda }_i,i=1,2,\dots ,N\right)$. Following Favaro and Walker (2010), we have the following approximation. $${\overline{G}}^{\ast n}\left(y\right)\approx \mathrm{Pr}\left(\sum _{j=1}^k{U}_j\le y\right)\times \mathrm{Pr}\left(y<\sum _{j=1}^{k+1}{U}_j\right).$$

By using the generalized hypergeometric series as shown below, (32) $${}_m{F}_n\left(a_1,\dots ,a_m;c_1,\dots ,c_n;d\right)=\sum _{k=0}^{\infty }\frac{{\left(a_1\right)}_k\dots {\left(a_m\right)}_k}{{\left(c_1\right)}_k\dots {\left(c_m\right)}_k}\frac{d^k}{k!},$$(32) as ${\left(a\right)}_0=1,$ ${\left(a\right)}_k=\left(a\right)\left(a+1\right)\dots \left(a+k-1\right)$ which is Pochhammer’s polynomial. Inserting this expansion into the expression of Favaro and Walker (2010, pp. 2041), we have the below approximation as, (33) $$\begin{array}{c}{\overline{G}}^{\ast n}\left(y\right)\approx {\left(\frac{1-e^{-y}}{e^{-y}}\right)}^b{\frac{\Gamma \left({\alpha }_k+{\beta }_k\right)}{\beta }}_2{F}_1\left({\alpha }_k+{\beta }_k,{\beta }_k;{\beta }_k+1;\frac{e^{-y}-1}{e^{-y}}\right)\\ \times (1-{\left(\frac{1-e^{-y}}{e^{-y}}\right)}^b\frac{\Gamma \left({\alpha }_{k+1}+{\beta }_{k+1}\right)}{{\beta }_{k+1}}{}_2{F}_1\left({\alpha }_{k+1}+{\beta }_{k+1},{\beta }_{k+1};{\beta }_{k+1}+1;\frac{e^{-y}-1}{e^{-y}}\right)),\end{array}$$(33) where ${\alpha }_{\left(s\right)}:=\gamma \left(s\right)/\left(\vartheta \left(s\right)-\gamma {\left(s\right)}^2\right),$ ${\beta }_{\left(s\right)}:=\left(1-\gamma \left(s\right)\right)\left(\gamma \left(s\right)-\vartheta \left(s\right)\right)/\left(\vartheta \left(s\right)-\gamma {\left(s\right)}^2\right)$, $\gamma \left(s\right):={\prod }_{i=1}^s{\lambda }_i\left({\lambda }_i+1\right)$, and $\vartheta \left(s\right):={\prod }_{i=1}^s{\lambda }_i\left({\lambda }_i+1\right)/\left({\lambda }_i+2\right).$ Then, by Equation (33)(33) $$\begin{array}{c}{\overline{G}}^{\ast n}\left(y\right)\approx {\left(\frac{1-e^{-y}}{e^{-y}}\right)}^b{\frac{\Gamma \left({\alpha }_k+{\beta }_k\right)}{\beta }}_2{F}_1\left({\alpha }_k+{\beta }_k,{\beta }_k;{\beta }_k+1;\frac{e^{-y}-1}{e^{-y}}\right)\\ \times (1-{\left(\frac{1-e^{-y}}{e^{-y}}\right)}^b\frac{\Gamma \left({\alpha }_{k+1}+{\beta }_{k+1}\right)}{{\beta }_{k+1}}{}_2{F}_1\left({\alpha }_{k+1}+{\beta }_{k+1},{\beta }_{k+1};{\beta }_{k+1}+1;\frac{e^{-y}-1}{e^{-y}}\right)),\end{array}$$(33) we get, (34) $$\begin{array}{c}{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{G}}^{\ast n}\left(u\right)du=\sum _{k=0}^{\infty }{\left(-1\right)}^k\frac{e^{-\xi \pi }\sqrt{\xi }}{k!}{\int }_t^{\infty }{u}^{c/2}{\overline{G}}^{\ast n}\left(u\right)du\\ =\sum _{k=0}^{\infty }{\left(-1\right)}^k\frac{e^{-\xi \pi }\sqrt{\xi }}{k!}\left[A_{\left(t\right)}-B_{\left(t\right)}\right],\end{array}$$(34) where, $$A_{\left(t\right)}={\int }_t^{\infty }{u}^{c/2}{\left(\frac{1-e^{-u}}{e^{-u}}\right)}^b{\frac{\Gamma \left({\alpha }_k+{\beta }_k\right)}{\beta }}_2{F}_1\left({\alpha }_k+{\beta }_k,{\beta }_k;{\beta }_k+1;\frac{e^{-u}-1}{e^{-u}}\right)du,$$ and $$\begin{array}{c}{B}_{\left(t\right)}=\frac{\Gamma \left({\alpha }_k+{\beta }_k\right)}{\beta }\frac{\Gamma \left({\alpha }_{k+1}+{\beta }_{k+1}\right)}{{\beta }_{k+1}}\\ \times {\int }_t^{\infty }{u}^{c/2}{\left(\frac{1-e^{-u}}{e^{-u}}\right)}^{2b}{}_2{F}_1\left({\alpha }_k+{\beta }_k,{\beta }_k;{\beta }_k+1;\frac{e^{-u}-1}{e^{-u}}\right)\\ {}_2{F}_1\left({\alpha }_{k+1}+{\beta }_{k+1},{\beta }_{k+1};{\beta }_{k+1}+1;\frac{e^{-u}-1}{e^{-u}}\right)du.\end{array}$$

By Equation (32)(32) $${}_m{F}_n\left(a_1,\dots ,a_m;c_1,\dots ,c_n;d\right)=\sum _{k=0}^{\infty }\frac{{\left(a_1\right)}_k\dots {\left(a_m\right)}_k}{{\left(c_1\right)}_k\dots {\left(c_m\right)}_k}\frac{d^k}{k!},$$(32) , we have, $$A_{\left(t\right)}=\frac{\Gamma \left({\alpha }_k+{\beta }_k\right)}{\beta }\sum _{i=0}^{\infty }\frac{{\left({\alpha }_k+{\beta }_k\right)}_i{\left({\beta }_k\right)}_i}{i!{\left({\beta }_k+1\right)}_i}{\int }_t^{\infty }{u}^{c/2}{\left(\frac{1-e^{-u}}{e^{-u}}\right)}^b{\left(\frac{e^{-u}-1}{e^{-u}}\right)}^{i}du.$$

With this binomial expansion and when, $$\begin{array}{c}\pi \left(t\right)={\int }_t^{\infty }{u}^{c/2}{e}^{u(l+r)}du\\ =-\frac{1}{r+1}\left[\begin{array}{c}{\left(-r-1\right)}^{-c/2}\Gamma \left(\frac{c+2}{2}\right)+t^{c/2}{\left(-t\left(1+r\right)\right)}^{-c/2}\\ \times \left(\Gamma \left(\frac{c+2}{2},-\left(r+1\right)t\right)-\Gamma \left(\frac{c+2}{2}\right)\right)\end{array}\right];r<-1,\end{array}$$ we get, $$A_{\left(t\right)}=\frac{\Gamma \left({\alpha }_k+{\beta }_k\right)}{\beta }\sum _{i=0}^{\infty }\frac{{\left({\alpha }_k+{\beta }_k\right)}_i{\left({\beta }_k\right)}_i}{i!{\left({\beta }_k+1\right)}_i}\sum _{l=0}^b\sum _{r=0}^i\left(\begin{array}{c}b\\ l\end{array}\right)\left(\begin{array}{c}i\\ r\end{array}\right){\left(-1\right)}^{r+b-l}\pi \left(t\right).$$

By using Pochhammer’s polynomial (${\left(a\right)}_k=\Gamma \left(a+k\right)/\Gamma \left(a\right)$), we have the following result. $$A_{\left(t\right)}=\frac{1}{\beta }\sum _{i=0}^{\infty }\frac{{\beta }_k\Gamma \left({\alpha }_k+{\beta }_k+i\right)}{i!\left({\beta }_k+i\right)}\sum _{l=0}^b\sum _{r=0}^i\left(\begin{array}{c}b\\ l\end{array}\right)\left(\begin{array}{c}i\\ r\end{array}\right){\left(-1\right)}^{r+b-l}\pi \left(t\right).$$

With similar steps, we can obtain B as follows. $$\begin{array}{c}{B}_{\left(t\right)}=\frac{\Gamma \left({\alpha }_k+{\beta }_k\right)}{\beta }\frac{\Gamma \left({\alpha }_{k+1}+{\beta }_{k+1}\right)}{{\beta }_{k+1}}\sum _{j=0}^{2b}\left(\begin{array}{c}2b\\ j\end{array}\right){\left(-1\right)}^{2b-j}\sum _{b=0}^{\infty }\frac{{\left({\alpha }_k+{\beta }_k\right)}_b{\left({\beta }_k\right)}_b}{b!{\left({\beta }_k+1\right)}_b}\\ \times \sum _{a=0}^{\infty }\frac{{\left({\alpha }_{k+1}+{\beta }_{k+1}\right)}_a{\left({\beta }_{k+1}\right)}_a}{a!{\left({\beta }_k+1\right)}_a}\sum _{l=0}^b\sum _{i=0}^a\left(\begin{array}{c}b\\ l\end{array}\right)\left(\begin{array}{c}a\\ i\end{array}\right){\left(-1\right)}^{l+i}\overline{\pi }\left(t\right),\end{array}$$ which is equivalent to, $$\begin{array}{c}{B}_{\left(t\right)}=\frac{1}{\beta {\beta }_{k+1}}\sum _{j=0}^{2b}\left(\begin{array}{c}2b\\ j\end{array}\right){\left(-1\right)}^{2b-j}\sum _{b=0}^{\infty }\frac{\Gamma \left({\alpha }_k+{\beta }_k+b\right)\Gamma \left({\beta }_k+b\right){\beta }_k}{b!\Gamma \left({\beta }_k+1+b\right)}\\ \times \sum _{a=0}^{\infty }\frac{\Gamma \left({\beta }_k+1\right)\Gamma \left({\alpha }_{k+1}+{\beta }_{k+1}+a\right)\Gamma \left({\beta }_{k+1}+a\right)}{a!\Gamma \left({\beta }_k+1+a\right)\Gamma \left({\beta }_{k+1}\right)}\sum _{l=0}^b\sum _{i=0}^a\left(\begin{array}{c}b\\ l\end{array}\right)\left(\begin{array}{c}a\\ i\end{array}\right){\left(-1\right)}^{l+i}\overline{\pi }\left(t\right),\end{array}$$ where, $$\begin{array}{c}\overline{\pi }\left(t\right)={\int }_t^{\infty }{u}^{c/2}{e}^{u(i+j+l)}du\\ =\frac{-1}{i+j+1}\left[\begin{array}{c}{\left(-i-j-1\right)}^{-c/2}\Gamma \left(\frac{c+2}{2}\right)+t^{c/2}{\left(-t\left(1+i+j\right)\right)}^{-c/2}\\ \times \left(\Gamma \left(\frac{c+2}{2},-\left(i+j+1\right)t\right)-\Gamma \left(\frac{c+2}{2}\right)\right)\end{array}\right];i+j<-1.\end{array}$$

Therefore, by Equations (34)(34) $$\begin{array}{c}{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{G}}^{\ast n}\left(u\right)du=\sum _{k=0}^{\infty }{\left(-1\right)}^k\frac{e^{-\xi \pi }\sqrt{\xi }}{k!}{\int }_t^{\infty }{u}^{c/2}{\overline{G}}^{\ast n}\left(u\right)du\\ =\sum _{k=0}^{\infty }{\left(-1\right)}^k\frac{e^{-\xi \pi }\sqrt{\xi }}{k!}\left[A_{\left(t\right)}-B_{\left(t\right)}\right],\end{array}$$(34) and (31)(31) $$\begin{array}{c}{\overline{K}}_{\mathbb{X}}\left(y\right)=\sum _{j=1}^{\infty }\left(1-\alpha \right)\left(1-\phi \right){\phi }^{j-1}{\overline{G}}^{\ast k}\left(y\right),\\ =\sum _{j=1}^{\infty }\Psi \left(j\right){\overline{G}}^{\ast k}\left(y\right),\end{array}$$(31) , we can conclude that, $$\begin{array}{c}{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{K}}_{\mathbb{X}}\left(u\right)du-c{e}^{-\xi \left(t^c+\pi \right)}{\overline{K}}_{\mathbb{X}}\left(t\right)\\ =\sum _{j=1}^{\infty }\Psi \left(j\right){\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{G}}^{\ast k}\left(u\right)du-\sum _{j=1}^{\infty }c\Psi \left(j\right)e^{-\xi \left(t^c+\pi \right)}{\overline{G}}^{\ast k}\left(t\right)\\ =\sum _{j=1}^{\infty }\sum _{k=0}^{\infty }{\left(-1\right)}^k\Psi \left(j\right)\frac{e^{-\xi \pi }\sqrt{\xi }}{k!}\left[A_{\left(t\right)}-B_{\left(t\right)}\right]-\sum _{j=1}^{\infty }c\Psi \left(j\right)e^{-\xi \left(t^c+\pi \right)}{\overline{G}}^{\ast k}\left(t\right)\\ =\sum _{j=1}^{\infty }\Psi \left(j\right)e^{-\xi \pi }\left[\sum _{k=0}^{\infty }{\left(-1\right)}^k\frac{\sqrt{\xi }}{k!}\left[A_{\left(t\right)}-B_{\left(t\right)}\right]-c{e}^{-\xi t^c}{\overline{G}}^{\ast k}\left(t\right)\right].\end{array}$$

Once $\Psi \left(j\right)$ is probability, then it is TP₂. In addition, we note that $e^{-\xi \pi }$ is TP₂; then, $\Psi \left(j\right)e^{-\xi \pi }$ is $T{P}_2.$ Therefore, following Karlin (1968, p. 21), and using the variation diminishing property, we obtain that ${\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{K}}_{\mathbb{X}}\left(u\right)du-c{e}^{-\xi \left(t^c+\pi \right)}{\overline{K}}_{\mathbb{X}}\left(t\right)$ has at most one change of sign for all t if one change occurs; it occurs from + to –. Hence, $$\frac{{\int }_t^{\infty }{e}^{-\xi \left(u^c+\pi \right)}{\overline{K}}_{\mathbb{X}}\left(u\right)du}{e^{-\xi \left(t^c+\pi \right)}{\overline{K}}_{\mathbb{X}}\left(t\right)}\text{is}\text{decreasing}\text{in}t.$$

The proof (ii) follows similar arguments.

Shock models have been studied by several authors, and such research has provided a realistic formulation for modeling certain reliability systems situated in random environments. Various models collected here are physically motivated. For instance, the extreme and cumulative shock models may be appropriate descriptions for the fracture of brittle materials, such as glass, and for damage due to the earthquakes or volcanic activity. In such cases, the time between two consecutive shocks (inter-arrival times), the damage caused by a shock, the system failure and the relationships among all these elements are modeled and characterized within a shock model. In the literature, two major types have been distinguished depending on whether or not the effect of the shock on the system is independent of its arrival time. We can assume independence between the shock effect and the arrival time; we can let N be the number of shocks survived by the system and $U_k$ be the random inter-arrival time between the $\left(k-1\right)-th$ and k – th shocks; and we can denote a sequence of surviving probabilities $\left\{Pr\left[N\ge k\right]=1-{\sum }_{i=0}^k\left(1-\alpha \right)\left(1-\phi \right){\phi }^{i-1}\right\},$ which is defined by the probability that the system still runs after the kth shock. The lifetime $X$ of the system is then given by $X={\sum }_{k=1}^N{U}_k.$ Therefore, this shock model represents the particular case of random sums. We can apply Theorem 9 to shock models where the interarrival times are exponentially distributed, where we consider that the survival function of the shock model can be written as follows. (35) $${\Delta }_{\mathbb{U}}\left(t\right)={\overline{K}}_{\mathbb{U}}\left(\Lambda \left(t\right)\right),$$(35) where $\Lambda \left(t\right)={\int }_0^t\lambda \left(u\right)du$ and ${\overline{K}}_{\mathbb{X}}\left(t\right),$ as described in Eq. (31)(31) $$\begin{array}{c}{\overline{K}}_{\mathbb{X}}\left(y\right)=\sum _{j=1}^{\infty }\left(1-\alpha \right)\left(1-\phi \right){\phi }^{j-1}{\overline{G}}^{\ast k}\left(y\right),\\ =\sum _{j=1}^{\infty }\Psi \left(j\right){\overline{G}}^{\ast k}\left(y\right),\end{array}$$(31) . Hence, we can apply Theorem 10 in a non-stationary model as follows.

Theorem 11.

Suppose that U is the lifetime of advice with survival function ${\Delta }_{\mathbb{U}}\left(t\right)$ given in Equation (35) and ${\rm Y}\left(t\right),$ $A_{\left(t\right)}$ and $B_{\left(t\right)},$ which is described in Equations (28) – (30). If $0<t<min\left[{\lambda }_i,i=0,1,\dots \right]$. Then, if ${\rm Y}\left(0\right)\ge (\le ){\rm Y}\left(t\right)$ for all t > 0 and if $\Lambda \left(t\right)$ is star-shaped (anti-star-shaped), therefore $U\in NB{U}_{GLt}\left[NW{U}_{GLt}\right].$

Now, we mention the closure properties of the IMIT class under Non-homogeneous Poisson shock models.

Proposition 8.

Let $\left[{\sum }_{s=0}^{k-1}\Psi \left(s\right)\right]/\Psi \left(k\right)$ be increasing (decreasing) in k, for all $k\ge 1,$ and let $\lambda \left(t\right)$ be decreasing in t, $t\in R^{+}$. Then, the lifetime U with survival function ${\Delta }_{\mathbb{U}}\left(t\right),$ defined as in Equation (35), satisfies the $U\in IG{r}_{\downarrow }$ $\left(DG{r}_{\downarrow }\right).$

Acknowledgements

The author would like to express their sincere gratitude to the anonymous editor, the associate editor, and the referees for their very constructive and valuable comments and suggestions that add to the quality of the manuscript and increase its readability.