Information Properties of Consecutive Systems using Fractional Generalized Cumulative Residual Entropy

1. Introduction

Over the past three decades, extensive research has focused on consecutive k-out-of-n systems and their variations. These models have been applied to various engineering contexts, including telecom microwave stations, oil pipelines, vacuum systems in electron accelerators, computer networks, telecommunications, engineering, and integrated circuit design. A consecutive k-out-of-n system can be classified by the arrangement of its components as either linear or circular, and by its functioning principle as either a failure or a good system. A linear consecutive k-out-of-n:G system comprises n independent and identically distributed (i.i.d.) components arranged linearly and is operational if and only if at least k consecutive components are functioning. The consecutive n-out-of-n:G system, which requires all n components to function, is equivalent to a classical series system. In contrast, the 1-out-of-n:G system, which needs at least one operational component, gives a system with parallel structure. Both the consecutive n-out-of-n:G and 1-out-of-n:G systems have been extensively studied in the literature under various assumptions and analytical frameworks. Comprehensive reviews of previous work on the topic are available in Jung and Kim [1], Shen and Zuo [2], Kuo and Zuo [3], Chang et al. [4], Boland and Samaniego [5], and Eryılmaz [6,7], along with their citations.

The derivation of the distribution of the lifetimes of a linear consecutive k-out-of-n:G system is plain in the case where $2k\ge n.$ This is because the survival function of the system’s lifetime can be formulated in terms of the probability of the union of disjoint events. In this case, the lifetime of each component is represented by $X_1,\dots ,X_n$ having a common probability density function (p.d.f) $f\left(x\right)$, cumulative distribution function (c.d.f) $F\left(x\right)$, and survival (reliability) function $S\left(x\right)=P(X>x)$. We denote the system’s lifetime by $T_{k|n:G}$. When $2k\ge n,$ Eryilmaz [8] have shown that the reliability function of the consecutive k-out-of-n:G system can be expressed as

$$S_{k|n:G}\left(x\right)=(n-k+1)S^k\left(x\right)-(n-k)S^{k+1}\left(x\right),x>0.$$

(1)

The concept of entropy is an important criterion for measuring the uncertainty of a random event. The Shannon differential entropy is defined by $H\left(X\right)=\mathbb{E}[-logf(X\left)\right],$ where “log" means for the natural logarithm, with convention $0log0=0.$ Various attempts have been made to define possible alternative information measures. To this aim, Rao et al. [9] introduced the concept of the cumulative residual entropy (CRE) as follows:

$$\mathcal{E}\left(X\right)=-{\int }_0^{\infty }S\left(x\right)logS\left(x\right)\mathrm{d}x={\int }_0^{\infty }S\left(x\right)\Lambda \left(x\right)\mathrm{d}x,$$

(2)

where

$$\Lambda \left(x\right)=-logS\left(x\right)={\int }_0^x\eta \left(u\right)\mathrm{d}u,x>0,$$

(3)

is the cumulative hazard function and $\eta \left(u\right)=f\left(u\right)/S\left(u\right),u>0,$ stands for the hazard rate function. In a recent development, Di Crescenzo et al. [10] have presented the fractional generalized cumulative residual entropy (FGCRE) as follows:

$${\mathcal{E}}_{\alpha }\left(X\right)={\displaystyle \frac{1}{\Gamma (\alpha +1)}}{\int }_0^{\infty }S\left(x\right){[-logS\left(x\right)]}^{\alpha }\mathrm{d}x={\int }_0^1{\displaystyle \frac{{\psi }_{\alpha }\left(u\right)}{f\left(F^{-1}\left(u\right)\right)}}du,$$

(4)

where

$${\psi }_{\alpha }\left(x\right)={\displaystyle \frac{x{(-logx)}^{\alpha }}{\Gamma (\alpha +1)}},0\le x\le 1$$

(5)

for all $\alpha \ge 0.$ We recall that related results about the FCRE (as a special case of the FGCRE) can be seen in Xiong et al. [11], Alomani and Kayid [12] and Kayid and Shrahili [13]. It is worth noting that if $\alpha$ is a positive integer, it can easily be seen that (4) becomes the measure of generalized CRE established by Psarrakos and Navarro [14].

The study of information properties in reliability systems and order statistics has been explored by several researchers in the literature. For example, Wong and Chen [15] showed that the difference between the average entropy of order statistics and the entropy of data distribution is a constant. They also showed that for symmetric distributions, the entropy of order statistics is symmetric about the median. Ebrahimi et al. [16] explored some properties of the Shannon entropy of the order statistics and showed that the Kullback–Leibler information functions involving order statistics are distribution-free. Toomaj and Doostparatst [17] obtained an expression for the Shannon differential entropy of coherent and mixed systems using the concept of system signature. Moreover, Toomaj and Doostparatst [18] have shown that the Kullback–Leibler information functions involving the lifetime of mixed systems and order statistics as well as the parent distribution are distribution-free. Furthermore, Toomaj et al. [19] leveraged the concept of system signature to analyze the CRE properties of mixed systems. Similarly, Alomani and Kayid [12] employed system signatures to investigate the fractional CRE of coherent systems. For a broader exploration of uncertainty measures in reliability systems, readers can refer to [13,20,21], and the cited references therein. Motivated by the established body of research on information measures in reliability, this paper delves into the uncertainty properties of CRE specifically within the framework of consecutive k-out-of-n systems. By building upon this foundation, we aim to contribute to a deeper understanding of FGCRE properties within this particular system configuration.

This paper is structured as goes after. In Section 2, we introduce a representation of the FGCRE for consecutive k-out-of-n systems with lifetime $T_{k|n:G}$ from a sample drawn from any c.d.f F. This representation is defined in terms of the FGCRE for consecutive k-out-of-n systems from a sample drawn from the uniform distribution. We then provide an in-depth analysis examining the preservation of stochastic ordering properties for this type of system. In the sequel part, we provide a number of bounds of the FGCRE of consecutive k-out-of-n systems. Several characterization results are achieved in Section 3. In Section 4, we present computational studies to validate and confirm the achieved outcomes. Specifically, we propose two nonparametric estimators for estimating the FGCRE of consecutive systems and demonstrate their application using both real and simulated data, emphasizing the potential practical value of these new estimators. In Section 5, the paper is concluded by presenting some essential points and, further, outlining some possible future investigations.

2. FGCRE of Consecutive k-out-of-n:G System

This section is organized into two key subsections. We first present a useful expression for the FGCRE of the lifetime of the consecutive k-out-of-n:G system. This analytical formulation serves as the foundation for the subsequent in-depth examination of the preservation of stochastic ordering properties inherent to this class of systems. In the second subsection, we establish and provide some useful bounds that offer remarkable utility in situations where the number of the components of the consecutive k-out-of-n:G systems is large.

2.1. Expression and Stochastic Orders

We now find an explicit expression for the FGCRE of the consecutive k-out-of-n:G system with lifetime $T_{k|n:G}$, where the component lifetimes have a common continuous c.d.f F. To do this, we use the probability integral transformation $U_{k|n:G}=F\left(T_{k|n:G}\right)$. It is known that the corresponding transformations of the system’s components, denoted as $U_i=F\left(X_i\right)$ for $i=1,\dots ,n$, are independent and identically distributed (i.i.d.) random variables (r.v.s) that follow a uniform distribution on the interval $[0,1]$. Using equation (1), when $2k\ge n$, the survival function of $U_{k|n:G}$ is given by

$${\overline{G}}_{k|n:G}\left(u\right)=(n-k+1){(1-u)}^k-(n-k){(1-u)}^{k+1},$$

(6)

for all $0<u<1.$ We now give the next theorem.

Theorem 1. 

For $2k\ge n,$ the FGCRE of $T_{k|n:G},$ can be expressed as follows:

$${\mathcal{E}}_{\alpha }\left(T_{k|n:G}\right)={\int }_0^1{\displaystyle \frac{{\psi }_{\alpha }\left({\overline{G}}_{k|n:G}\left(u\right)\right)}{f\left(F^{-1}\left(u\right)\right)}}du,$$

(7)

where ${\psi }_{\alpha }\left(x\right)$ and ${\overline{G}}_{k|n:G}\left(u\right)$ are defined in (5) and (6), respectively.

Proof. 

Applying the transformation $u=F\left(x\right)$ and referring to (1) and (4), we get

$$\begin{array}{ccc}\hfill {\mathcal{E}}_{\alpha }\left(T_{k|n:G}\right)& =& {\displaystyle \frac{1}{\Gamma (\alpha +1)}}{\int }_0^{\infty }{S}_{k|n:G}\left(x\right){[-log{S}_{k|n:G}\left(x\right)]}^{\alpha }dx\hfill \\ & =& {\int }_0^{\infty }{\psi }_{\alpha }\left(S_{k|n:G}\left(x\right)\right)dx\hfill \\ & =& {\int }_0^{\infty }{\psi }_{\alpha }((n-k+1)S^k\left(x\right)-(n-k)S^{k+1}\left(x\right))dx\hfill \\ & =& {\int }_0^1{\displaystyle \frac{{\psi }_{\alpha }((n-k+1){(1-u)}^k-(n-k){(1-u)}^{k+1})}{f\left(F^{-1}\left(u\right)\right)}}du\hfill \end{array}$$

(8)

$$\begin{array}{ccc}& =& {\int }_0^1{\displaystyle \frac{{\psi }_{\alpha }\left({\overline{G}}_{k|n:G}\left(u\right)\right)}{f\left(F^{-1}\left(u\right)\right)}}du,\hfill \end{array}$$

(9)

and this completes the proof. □

The following example demonstrates the application of equation (7) in the consecutive k-out-of-n:G system.

Example 1. 

Let us consider a linear consecutive 3-out-of-5:G system with a lifetime

$$T_{3|5:G}=max(min(X_1,X_2,X_3),min(X_2,X_3,X_4),min(,X_3,X_4,X_5)).$$

Assume further that the lifetimes of the components are i.i.d. following the common Lomax distribution, also known as the Pareto Type II distribution. The p.d.f of the Lomax distribution is given by

$$f\left(x\right)={\displaystyle \frac{2}{\lambda }}{\left(1+{\displaystyle \frac{x}{\lambda }}\right)}^{-3},x>0,$$

(10)

where $\lambda >0$ denotes the scale parameters. It is clear that $f\left(F^{-1}\left(u\right)\right)={\displaystyle \frac{2}{\lambda }}{(1-u)}^{{\displaystyle \frac{3}{2}}}$ for all $0<u<1$. Through algebraic manipulations, we can derive the following expression

$$\begin{array}{c}\hfill {\mathcal{E}}_{\alpha }\left(T_{3|5:G}\right)={\displaystyle \frac{\lambda }{2}}{\int }_0^1{\psi }_{\alpha }\left({\overline{G}}_{k|n:G}\left(u\right)\right){(1-u)}^{-{\displaystyle \frac{3}{2}}}du.\end{array}$$

(11)

It is clear that the FGCRE is an increasing function of the scale parameter $\lambda$ for all $\alpha \ge 0.$ This means that as the scale parameter $\lambda$ increases, the system’s uncertainty concerning the FGCRE increases. So, this gives the significant impact of the Lomax distribution with the scale parameter $\lambda$ on the FGCRE, and thus the uncertainty of the system lifetime.

We now demonstrate that the FGCRE of the consecutive k-out-of-n:G systems is preserved under the dispersive and location-independent riskier orders. To begin with, let us review some stochastic orders and aging notion concepts. Hereafter, we denote the set of absolutely continuous nonnegative r.v.s with support $(0,\infty )$ as ${\mathfrak{R}}_{+}=\{X;X\ge 0\}.$

Definition 1. 

Let $X,Y\in {\mathfrak{R}}_{+}$ be r.v.s with pdfs $f_X\left(x\right)$ and $f_Y\left(x\right),$ cdfs $F_X\left(x\right)$ and $F_Y\left(x\right),$ survival functions $S_X\left(x\right)$ and $S_Y\left(x\right)$ and hazard rate orders ${\lambda }_X\left(x\right)=f_X\left(x\right)/S_X\left(x\right)$ and ${\lambda }_Y\left(x\right)=f_Y\left(x\right)/S_Y\left(x\right),$ respectively. Then, we say that

1. 

X has decreasing failure rate (DFR) if ${\lambda }_X\left(x\right)$ is decreasing in $x;$

2. 

X is smaller than Y in the hazard rate order (denoted by $X{\le }_{hr}Y$) if ${\lambda }_X\left(x\right)\ge {\lambda }_Y\left(x\right)$ for all $x>0;$

3. 

X is smaller than Y in the dispersive order (denoted by $X{\le }_{d}Y$) if $F_X^{-1}\left(v\right)-F_X^{-1}\left(u\right)\le F_Y^{-1}\left(v\right)-F_Y^{-1}\left(u\right),0<u\le v<1;$

4. 

X is smaller than Y in the location-independent riskier order (denoted by $X{\le }_{lir}Y$) if ${\int }_0^{F_X^{-1}\left(p\right)}{F}_X\left(x\right)dx\le {\int }_0^{F_Y^{-1}\left(p\right)}{F}_Y\left(x\right)dx,p\in (0,1);$

The dispersive ${\le }_d$ order was initially explored by Bickel and Lehmann [22] in nonparametric statistics, while the ${\le }_{lir}$ order was proposed by Jewitt [23] in expected utility theory and its applications in insurance. We recall that $X{\le }_{d}Y$ is equivalent to

$$f_Y\left(F_Y^{-1}\left(v\right)\right)\le f_X\left(F_X^{-1}\left(v\right)\right),0<v<1.$$

(12)

Additionally, the following implications are well-recognized:

$$\begin{array}{ccc}\hfill \mathrm{if}X{\le }_{hr}Y\mathrm{and}\mathrm{either}X\mathrm{or}Y\mathrm{is}\mathrm{DFR}& ⟹& X{\le }_{d}Y⟹X{\le }_{lir}Y.\hfill \end{array}$$

(13)

Since ${\psi }_{\alpha }\left(x\right)\ge 0$ for all $0\le x\le 1$ and $\alpha \ge 0$, relations (4) and (12) imply that ${\mathcal{E}}_{\alpha }\left(X\right)\le {\mathcal{E}}_{\alpha }\left(Y\right)$ when $X{\le }_{d}Y$. Consequently, from implication (13), we obtain the following corollary.

Corollary 1. 

If $X{\le }_{hr}Y$ and either X or Y is DFR, then ${\mathcal{E}}_{\alpha }\left(X\right)\le {\mathcal{E}}_{\alpha }\left(Y\right).$

Consider Z as a r.v. with c.d.f H given by:

$${\eta }_Z\left(z\right)={\int }_0^{z}H\left(v\right)dv,z>0.$$

(14)

Landsberger and Meilijson [24] showed that

$$X{\le }_{lir}Y⟺{\eta }_Y^{-1}\left(z\right)-{\eta }_X^{-1}\left(z\right)\mathrm{is}\mathrm{increasing}\mathrm{in}z>0.$$

(15)

Here, we present a theorem demonstrating that the FGCRE of a series system with k components is lower than that of a consecutive k-out-of-n:G system, assuming both systems’ components exhibit the DFR property.

Theorem 2. 

For $2k\ge n,$ let $T_{k|n:G}$ be the lifetime of consecutive k-out-of-n:G system having the common p.d.f $f_X\left(x\right)$ and c.d.f $F_X\left(x\right).$ If X is DFR, then ${\mathcal{E}}_{\alpha }\left(X_{1:k}\right)\le {\mathcal{E}}_{\alpha }\left(T_{k|n:G}\right)$ for all $\alpha \ge 0.$

Proof. 

Since X is DFR, $X_{1:k}$ is also DFR. By applying Theorem 4.5 from Eryılmaz and Navarro [25], we have $X_{1:k}{\le }_{hr}{T}_{k|n:G}$. Therefore, Corollary 1 completes the proof. □

The next theorem outlines the conditions for preserving the dispersive order in consecutive systems. The proof is omitted as it is deemed straightforward.

Theorem 3. 

For $2k\ge n,$ let $T_{k|n:G}^X$ and $T_{k|n:G}^Y$ be the lifetimes of two consecutive k-out-of-n:G systems having the common pdfs $f_X\left(x\right)$ and $f_Y\left(x\right)$ and cdfs $F_X\left(x\right)$ and $F_Y\left(x\right),$ respectively. If $X{\le }_{d}Y,$ then ${\mathcal{E}}_{\alpha }\left(T_{k|n:G}^X\right)\le {\mathcal{E}}_{\alpha }\left(T_{k|n:G}^Y\right)$ for all $\alpha \ge 0.$

The following example illustrates the application of Theorem 3.

Example 2. 

Consider two consecutive k-out-of-n:G systems with lifetimes $T_{k|n:G}^X$ and $T_{k|n:G}^Y,$ respectively. The system $T_{k|n:G}^X$ have i.i.d. component lifetimes $X_1,X_2,\dots ,X_n$, which follow a Makeham distribution with the survival function $S\left(x\right)=e^{-x+a(x+e^{-x}-1)}$, where $x>0$ and $a>0$. Moreover, the system $T_{k|n:G}^Y$ have i.i.d. component lifetimes $Y_1,Y_2,\dots ,Y_n$ that follow an exponential distribution with the survival function $S_Y\left(x\right)=e^{-x},x>0$. It is clear that ${\lambda }_X\left(x\right)=1+a(1-e^{-x})$ and ${\lambda }_Y\left(x\right)=1$. Comparing the hazard rate functions shows that ${\lambda }_X\left(x\right)>{\lambda }_Y\left(x\right)$ for $a>0$, indicating that $X{\le }_{hr}Y$. Given that Y has the DFR property, relation (13) leads to $X{\le }_{d}Y$. Consequently, by Corollary 3, we have ${\mathcal{E}}_{\alpha }\left(T_{k|n:G}^X\right)\le {\mathcal{E}}_{\alpha }\left(T_{k|n:G}^Y\right)$ for all $\alpha \ge 0.$ This indicates that the uncertainty of the system with lifetime $T_{k|n:G}^X$ is less than or equal to that of the system with lifetime $T_{k|n:G}^Y$, according to the FGCRE measure.

The next theorem outlines the conditions for preserving location-independent riskier orders in the formation of consecutive systems.

Theorem 4. 

In the setting of Theorem 3 let ${\varphi }_{k|n}\left(t\right)=(n-k+1){(1-t)}^k-(n-k){(1-t)}^{k+1},0<t<1.$ For $2k\ge n,$ if $X{\le }_{lir}Y$ and

$${\displaystyle \frac{{\psi }_{\alpha }\left({\varphi }_{k|n}\left(t\right)\right)}{t}},0\le t\le 1,$$

(16)

is a decreasing function of t, then ${\mathcal{E}}_{\alpha }\left(T_{k|n:G}^X\right)\le {\mathcal{E}}_{\alpha }\left(T_{k|n:G}^Y\right)$ for all $\alpha \ge 0$.

Proof. 

It is clear that Eq. (1) can be represented as $S_{k|n:G}\left(x\right)={\varphi }_{k|n}\left(F\left(x\right)\right).$ From (15), we derive

$${\displaystyle \frac{d}{dz}}({\eta }_Y^{-1}\left(z\right)-{\eta }_X^{-1}\left(z\right))={\displaystyle \frac{1}{F_Y\left({\eta }_Y^{-1}\left(z\right)\right)}}-{\displaystyle \frac{1}{F_X\left({\eta }_X^{-1}\left(z\right)\right)}}\ge 0,$$

which implies

$$F_X\left(z\right)\ge F_Y\left({\eta }_Y^{-1}\left({\eta }_X\left(z\right)\right)\right),$$

(17)

for all $z>0.$ From relations (1) and (4), we obtain

$$\begin{array}{ccc}\hfill {\int }_0^{\infty }{\displaystyle \frac{S_{k|n:G}^X\left(x\right){[-log{S}_{k|n:G}^X\left(x\right)]}^{\alpha }}{\Gamma (\alpha +1)}}dx& =& {\int }_0^{\infty }{\psi }_{\alpha }\left(S_{k|n:G}^X\left(x\right)\right)dx\hfill \\ & =& {\int }_0^{\infty }{\displaystyle \frac{{\psi }_{\alpha }\left(S_{k|n:G}^X\left(x\right)\right)}{F_X\left(x\right)}}{F}_X\left(x\right)\mathrm{d}x\hfill \\ & =& {\int }_0^{\infty }{\displaystyle \frac{{\psi }_{\alpha }\left({\varphi }_{k|n}\left(F_X\left(x\right)\right)\right)}{F_X\left(x\right)}}{F}_X\left(x\right)\mathrm{d}x\hfill \\ & \le & {\int }_0^{\infty }{\displaystyle \frac{{\psi }_{\alpha }\left({\varphi }_{k|n}\left(F_Y\left({\eta }_Y^{-1}\left({\eta }_X\left(x\right)\right)\right)\right)\right)}{F_Y\left({\eta }_Y^{-1}\left({\eta }_X\left(x\right)\right)\right)}}{F}_X\left(x\right)\mathrm{d}x.\hfill \end{array}$$

(18)

The inequality arises from Eq. (17) and using the fact that ${\psi }_{\alpha }\left({\varphi }_{k|n}\left(t\right)\right)/t$ is a decreasing function of $0\le t\le 1.$ Furthermore, by setting $u={\eta }_Y^{-1}\left({\eta }_X\left(x\right)\right)$, we have

$$dx={\displaystyle \frac{F_Y\left(u\right)}{F_X\left({\eta }_X^{-1}\left({\eta }_Y\left(u\right)\right)\right)}}du.$$

Upon using this, (18) reduces to

$$\begin{array}{ccc}& & {\int }_{{\eta }_Y^{-1}\left({\eta }_X\left(0\right)\right)}^{\infty }{\displaystyle \frac{{\psi }_{\alpha }\left({\varphi }_{k|n}\left(F_Y\left(u\right)\right)\right)}{F_Y\left(u\right)}}{\displaystyle \frac{F_X\left({\eta }_X^{-1}\left({\eta }_Y\left(u\right)\right)\right)F_Y\left(u\right)}{F_X\left({\eta }_X^{-1}\left({\eta }_Y\left(u\right)\right)\right)}}\mathrm{d}u\hfill \\ & =& {\int }_{{\eta }_Y^{-1}\left({\eta }_X\left(0\right)\right)}^{\infty }{\psi }_{\alpha }\left({\varphi }_{k|n}\left(F_Y\left(u\right)\right)\right)\mathrm{d}u\hfill \\ & =& {\int }_0^{\infty }{\psi }_{\alpha }\left(S_{k|n:G}^Y\left(x\right)\right)dx\hfill \\ & =& {\int }_0^{\infty }{\displaystyle \frac{S_{k|n:G}^Y\left(x\right){[-log{S}_{k|n:G}^Y\left(x\right)]}^{\alpha }}{\Gamma (\alpha +1)}}dx.\hfill \end{array}$$

The final equality in the previous relation follows from the fact that ${\eta }_Y^{-1}\left({\eta }_X\left(0\right)\right)=0$, implying that ${\mathcal{E}}_{\alpha }\left(T_{k|n:G}^X\right)\le {\mathcal{E}}_{\alpha }\left(T_{k|n:G}^Y\right).$ Hence, the theorem. □

2.2. Some Bounds

Due to the absence of closed-form expressions for the FGCRE of consecutive systems across various distributions and with numerous components, it is essential to employ bounding techniques to estimate the FGCRE of the system’s lifetime. Recognizing this challenge, we aim to explore the effectiveness of these bounds in characterizing the FGCRE of consecutive systems. The initial finding establishes a bound on the system’s FGCRE based on the common FGCRE of its components.

Theorem 5. 

For $2k\ge n,$ the FGCRE of $T_{k|n:G}$ are bounded as follows:

$$\begin{array}{c}\hfill {\mathfrak{B}}_{\alpha }{\mathcal{E}}_{\alpha }\left(X_1\right)\le {\mathcal{E}}_{\alpha }\left(T_{k|n:G}\right)\le {\mathfrak{D}}_{\alpha }{\mathcal{E}}_{\alpha }\left(X_1\right),\end{array}$$

where ${\mathfrak{B}}_{\alpha }={inf}_{u\in (0,1)}{\displaystyle \frac{{\psi }_{\alpha }\left({\overline{G}}_{k|n:G}\left(u\right)\right)}{{\psi }_{\alpha }\left(u\right)}}$, ${\mathfrak{D}}_{\alpha }={sup}_{u\in (0,1)}{\displaystyle \frac{{\psi }_{\alpha }\left({\overline{G}}_{k|n:G}\left(u\right)\right)}{{\psi }_{\alpha }\left(u\right)}}.$

Proof. 

The upper bound can be identified from (7) as follows:

$$\begin{array}{cc}\hfill {\mathcal{E}}_{\alpha }\left(T_{k|n:G}\right)& ={\int }_0^1{\displaystyle \frac{{\psi }_{\alpha }\left({\overline{G}}_{k|n:G}\left(u\right)\right)}{f\left(F^{-1}\left(u\right)\right)}}du={\int }_0^1{\displaystyle \frac{{\psi }_{\alpha }\left({\overline{G}}_{k|n:G}\left(u\right)\right)}{{\psi }_{\alpha }\left(u\right)}}{\displaystyle \frac{{\psi }_{\alpha }\left(u\right)}{f\left(F^{-1}\left(u\right)\right)}}du\hfill \\ \hfill & \le \underset{u\in (0,1)}{sup}{\displaystyle \frac{{\psi }_{\alpha }\left({\overline{G}}_{k|n:G}\left(u\right)\right)}{{\psi }_{\alpha }\left(u\right)}}{\int }_0^1{\displaystyle \frac{{\psi }_{\alpha }\left(u\right)}{f\left(F^{-1}\left(u\right)\right)}}du={\mathfrak{D}}_{\alpha }{\mathcal{E}}_{\alpha }\left(X_1\right).\hfill \end{array}$$

The lower bound can be similarly derived. □

The upcoming theorem introduces additional useful bounds based on the extremes of the p.d.f and the function ${\psi }_{\alpha }\left(u\right).$

Theorem 6. 

Let $T_{k|n:G}$ be the lifetime of consecutive k-out-of-n:G system having the common pdfs $f_X\left(x\right)$ and c.d.f $F_X\left(x\right).$ If S is the support of f, $m={inf}_{x\in S}f\left(x\right)$ and $M={sup}_{x\in S}f\left(x\right)$, then

$$\begin{array}{c}\hfill {\displaystyle \frac{{\mathcal{E}}_{\alpha }\left(U_{k|n:G}\right)}{M}}\le {\mathcal{E}}_{\alpha }\left(T_{k|n:G}\right)\le {\displaystyle \frac{{\mathcal{E}}_{\alpha }\left(U_{k|n:G}\right)}{m}},\end{array}$$

(19)

where ${\mathcal{E}}_{\alpha }\left(U_{k|n:G}\right)={\int }_0^1{\psi }_{\alpha }\left({\overline{G}}_{k|n:G}\left(u\right)\right)du.$

Proof. 

Since $m\le f\left(F^{-1}\left(u\right)\right)\le M,0<u<1$, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{M}}{\int }_0^1{\psi }_{\alpha }\left({\overline{G}}_{k|n:G}\left(u\right)\right)du\le {\int }_0^1{\displaystyle \frac{{\psi }_{\alpha }\left({\overline{G}}_{k|n:G}\left(u\right)\right)}{f\left(F^{-1}\left(u\right)\right)}}du\le {\displaystyle \frac{1}{m}}{\int }_0^1{\psi }_{\alpha }\left({\overline{G}}_{k|n:G}\left(u\right)\right)du.\end{array}$$

Upon recalling Eq. (7), we have the result. □

The term ${\mathcal{E}}_{\alpha }\left(U_{k|n:G}\right)$ denotes the FGCRE of a consecutive k-out-of-n:G system with a uniform distribution on the interval $(0,1)$. The bounds in Eq. (19) depend on the extremes of the p.d.f f. If the lower bound m is zero, an upper bound does not exist. Conversely, if the upper bound M is infinite, a lower bound is absent. The following example illustrates the application of the bounds from Theorems 5 and 6 for a consecutive k-out-of-n:G system.

Example 3. 

Consider a linear consecutive 6-out-of-10:G system. The system lifetime, denoted by The system lifetime, denoted by $T_{6|10:G}$, is defined as the maximum of the minimum values within consecutive blocks of nine components. Specifically, $T_{6|10:G}=max(X_{[1:6]},X_{[2:7]},\dots ,X_{[6:10]}),$ where $X_{[j:r]}=min(X_j,\dots ,X_r)$ for $1\le j<r\le 10.$ Assuming an exponential distribution with mean $\mu$, it is easy to see that $m=0$ and $M=1/\mu .$ Moreover, one can see that ${\mathfrak{B}}_{\alpha }=0$ for all $\alpha \ge 0.$ On the other hand, one can obtain ${\mathcal{E}}_{\alpha }\left(X_1\right)=\mu$ for all $\alpha \ge 0.$ Through algebraic manipulations, we can derive the following expression

$$\begin{array}{c}\hfill {\mathcal{E}}_{\alpha }\left(T_{6|10:G}\right)=\mu {\int }_0^1{\displaystyle \frac{{\psi }_{\alpha }\left({\overline{G}}_{k|n:G}\left(u\right)\right)}{(1-u)}}du,\end{array}$$

(20)

for all $\alpha \ge 0.$ It is not easy to come up with an exact expression for the value and given bounds, so we have to use numerical computations. In Table 1, we listed the values of these expressions. The bounds mentioned in Theorem 6 are important, easy and useful for the application.

3. Characterization Results

The aim of this section is to present some characterization results using the FGCRE properties of consecutive k-out-of-n:G systems. To start, we need the following lemma which is a direct corollary of the Stone–Weierstrass Theorem (see Aliprantis and Burkinshaw, [26]). The given lemma will be used to prove the main results in this subsection.

Lemma 1. 

If ζ is a continuous function on $[0,1]$ such that ${\int }_0^1{x}^n\zeta \left(x\right)dx=0$ for all $n\ge 0,$ then $\zeta \left(x\right)=0$ for any $x\in [0,1].$

Building upon this foundational result, we can establish that the parent distribution of a lifetime r.v. can be uniquely characterized by the FGCRE of $T_{k|n:G},$ where ${\left\{k_j\right\}}_{j\ge 1}$ is a strictly increasing sequence of positive integers.

Theorem 7. 

Let $T_{k|n:G}^X$ and $T_{k|n:G}^Y$ be lifetimes of two consecutive k-out-of-n:G systems having the common pdfs $f_X\left(x\right)$ and $f_Y\left(x\right)$ and cdfs $F_X\left(x\right)$ and $F_Y\left(x\right),$ respectively. Then $F_X$ and $F_Y$ are members of one family of distributions if, and only if, for a fixed $n,$

$${\mathcal{E}}_{\alpha }\left(T_{k|n:G}^X\right)={\mathcal{E}}_{\alpha }\left(T_{k|n:G}^Y\right),$$

(21)

for all $2k\ge n.$

Proof. 

The necessity is trivial and therefore we need to prove the sufficiency part. First note that Eq. (7) can be rewritten as follows:

$$\begin{array}{ccc}\hfill {\mathcal{E}}_{\alpha }\left(T_{k|n:G}^X\right)& =& {\int }_0^1{\displaystyle \frac{{\psi }_{\alpha }\left({\overline{G}}_{k|n:G}\left(u\right)\right)}{f_X\left(F_X^{-1}\left(u\right)\right)}}du\hfill \\ & =& -{\int }_0^1{(1-u)}^{2k-n}{\displaystyle \frac{{(1-u)}^{n-k}((n-k+1)-(n-k)(1-u))log\left({\overline{G}}_{k|n:G}\left(u\right)\right)}{f_X\left(F_X^{-1}\left(u\right)\right)}}du\hfill \\ & =& {\int }_0^1{z}^{2k-n}{\displaystyle \frac{\Omega \left(z\right)}{f_X\left(F_X^{-1}(1-z)\right)}}dz,(\mathrm{taking}z=1-u)\hfill \end{array}$$

(22)

where $\Omega \left(z\right)=-z^{n-k}((n-k+1)-(n-k)z)log\left({\overline{G}}_{k|n:G}\left(z\right)\right),0<z<1.$ Similar argument holds for ${\mathcal{E}}_{\alpha }\left(T_{k|n:G}^Y\right).$ From (22), we have

$${\int }_0^1\left[{\displaystyle \frac{1}{f_X\left(F_X^{-1}(1-z)\right)}}-{\displaystyle \frac{1}{f_Y\left(F_Y^{-1}(1-z)\right)}}\right]z^k\Omega \left(z\right)dz=0.$$

According to Lemma 1, we can conclude that

$$f_X\left(F_X^{-1}(1-z)\right)=f_Y\left(F_Y^{-1}(1-z)\right),a.e.z\in (0,1),$$

or equivalently $f_X\left(F_X^{-1}\left(v\right)\right)=f_Y\left(F_Y^{-1}\left(v\right)\right),0<v<1.$ So, it follows that $F_X^{-1}\left(v\right)=F_Y^{-1}\left(v\right)+d$ where d is a constant. By noting that ${lim}_{v\to 0}{F}_X^{-1}\left(v\right)={lim}_{v\to 0}{F}_Y^{-1}\left(v\right)=0$ for all $v\in (0,1),$ we have $F_X^{-1}\left(v\right)=F_Y^{-1}\left(v\right).$ This relationship implies that $F_X$ and $F_Y$ are members of one family of distributions. □

Since a consecutive n-out-of-n:G system is a series system (as mentioned before), the next corollary details its characteristics.

Corollary 2. 

Under the conditions of Theorem 7, $F_X$ and $F_Y$ are members of one family of distributions if, and only if,

$${\mathcal{E}}_{\alpha }\left(T_{n|n:G}^X\right)={\mathcal{E}}_{\alpha }\left(T_{n|n:G}^Y\right),$$

for all $n\ge 1.$

A further characterization is established in the following theorem.

Theorem 8. 

In the setting of Theorem 7, $F_X$ and $F_Y$ are members of one family of distributions with a change in the scale if, and only if, for a fixed $n,$

$${\displaystyle \frac{{\mathcal{E}}_{\alpha }\left(T_{k|n:G}^X\right)}{{\mathcal{E}}_{\alpha }\left(X\right)}}={\displaystyle \frac{{\mathcal{E}}_{\alpha }\left(T_{k|n:G}^Y\right)}{{\mathcal{E}}_{\alpha }\left(Y\right)}},$$

(23)

for all $2k\ge n.$

Proof. 

The necessity is trivial and hence it remains to prove the sufficiency part. From (22), we can write

$${\displaystyle \frac{{\mathcal{E}}_{\alpha }\left(T_{k|n:G}^X\right)}{{\mathcal{E}}_{\alpha }\left(X\right)}}={\int }_0^1{z}^{2k-n}{\displaystyle \frac{\Omega \left(z\right)}{{\mathcal{E}}_{\alpha }\left(X\right)f_X\left(F_X^{-1}(1-z)\right)}}dz.$$

(24)

Similar argument can be obtained for ${\mathcal{E}}_{\alpha }\left(T_{k|n:G}^Y\right)/{\mathcal{E}}_{\alpha }\left(Y\right).$ From (23) and (24), we obtain

$${\int }_0^1{z}^{2k-n}{\displaystyle \frac{\Omega \left(z\right)}{{\mathcal{E}}_{\alpha }\left(X\right)f_X\left(F_X^{-1}(1-z)\right)}}dz={\int }_0^1{z}^{2k-n}{\displaystyle \frac{\Omega \left(z\right)}{{\mathcal{E}}_{\alpha }\left(Y\right)f_Y\left(F_Y^{-1}(1-z)\right)}}dz.$$

(25)

Let us set $c={\mathcal{E}}_{\alpha }\left(Y\right)/{\mathcal{E}}_{\alpha }\left(X\right).$ Then, (25) can be expressed as

$${\int }_0^1\left[{\displaystyle \frac{1}{f_X\left(F_X^{-1}(1-z)\right)}}-{\displaystyle \frac{1}{c{f}_Y\left(F_Y^{-1}(1-z)\right)}}\right]z^{2k-n}\Omega \left(z\right)dz=0.$$

Applying Lemma 1, we can conclude that

$$f_X\left(F_X^{-1}(1-z)\right)=c{f}_Y\left(F_Y^{-1}(1-z)\right),a.e.z\in (0,1),$$

or equivalently $f_X\left(F_X^{-1}\left(v\right)\right)=c{f}_Y\left(F_Y^{-1}\left(v\right)\right),0<v<1.$ Given the preceding discussion, it follows that $F_X^{-1}\left(v\right)=c{F}_Y^{-1}\left(v\right)+d.$ By noting that ${lim}_{v\to 0}{F}_X^{-1}\left(v\right)={lim}_{v\to 0}{F}_Y^{-1}\left(v\right)=0$ for all $v\in (0,1),$ we have $F_X^{-1}\left(v\right)=c{F}_Y^{-1}\left(v\right).$ This means that $F_X$ and $F_Y$ are members of one family of distributions with a change in the scale. □

Using Theorem 8, we get the following corollary.

Corollary 3. 

Under the assumptions given in Theorem 8, $F_X$ and $F_Y$ are members of one family of distributions with a change in the scale if, and only if,

$${\displaystyle \frac{{\mathcal{E}}_{\alpha }\left(T_{n|n:G}^X\right)}{{\mathcal{E}}_{\alpha }\left(X\right)}}={\displaystyle \frac{{\mathcal{E}}_{\alpha }\left(T_{n|n:G}^Y\right)}{{\mathcal{E}}_{\alpha }\left(Y\right)}},$$

for all $n\ge 1.$

4. Nonparametric Estimation

Hereafter, we present two nonparametric estimators of the FGCRE of consecutive k-out-of-n:G systems. To do this, let us consider a sequence of absolutely continuous, non-negative, i.i.d. r.v.s $X_1,X_2,\dots ,X_N$, where $X_{1:N}\le X_{2:N}\le \cdots \le X_{N:N}$ denote their order statistics. By recalling (7), the FGCRE of $T_{k|n:G}$ can be rewritten as

$$\begin{array}{ccc}\hfill {\mathcal{E}}_{\alpha }\left(T_{k|n:G}\right)& =& {\int }_0^1{\displaystyle \frac{{\psi }_{\alpha }\left({\overline{G}}_{k|n:G}\left(u\right)\right)}{f\left(F^{-1}\left(u\right)\right)}}du={\int }_0^1{\psi }_{\alpha }\left({\overline{G}}_{k|n:G}\left(u\right)\right)\left[{\displaystyle \frac{d{F}^{-1}\left(u\right)}{du}}\right]du\hfill \\ & =& {\int }_0^1{\psi }_{\alpha }((n-k+1){(1-u)}^k-(n-k){(1-u)}^{k+1})\left[{\displaystyle \frac{d{F}^{-1}\left(u\right)}{du}}\right]du,\hfill \end{array}$$

(26)

which holds for $2k\ge n.$ Based on (26), we estimate the FGCRE ${\mathcal{E}}_{\alpha }\left(T_{k|n:G}\right).$ This approach uses derivative estimates of the inverse distribution function at the sample points. Following Vasicek [27], we estimate the derivative ${\displaystyle \frac{d{F}^{-1}\left(u\right)}{du}}$ by approximating it as the slope, defined as:

$${\displaystyle \frac{d{F}^{-1}\left(u\right)}{du}}={\displaystyle \frac{N(X_{i+m:N}-X_{i-m:N})}{2m}},$$

where $X_{i:N}=X_{1:N}$ for $i<1$ and $X_{i:N}=X_{N:N}$ for $i>N.$ In this case, N is the sample size and m is a positive integer referred to as the window size, satisfying $m\le N/2.$ This allows us to obtain the first estimator for ${\mathcal{E}}_{\alpha }\left(T_{k|n:G}\right)$ given by

$$\begin{array}{ccc}\hfill {\widehat{\mathcal{E}}}_{\alpha ,1}\left(T_{k|n:G}\right)& =& {\displaystyle \frac{1}{N}}\sum _{i=1}^N{\psi }_{\alpha }\left({\overline{G}}_{k|n:G}\left({\displaystyle \frac{i}{N+1}}\right)\right)\left({\displaystyle \frac{N(X_{i+m:N}-X_{i-m:N})}{2m}}\right)\hfill \\ & =& {\displaystyle \frac{1}{N}}\sum _{i=1}^N{\psi }_{\alpha }\left((n-k+1){\left(1-{\displaystyle \frac{i}{N+1}}\right)}^k-(n-k){\left(1-{\displaystyle \frac{i}{N+1}}\right)}^{k+1}\right)\hfill \\ & \times & \left({\displaystyle \frac{N(X_{i+m:N}-X_{i-m:N})}{2m}}\right),\hfill \end{array}$$

(27)

for all $2k\ge n$ and $\alpha \ge 0.$ We now introduce the second estimator based on the empirical survival function for the sample corresponding to $S\left(x\right)$ as

$$S_N\left(x\right)=\sum _{i=1}^{N-1}{\displaystyle \frac{N-i}{N}}{I}_{[X_{i:N},x_{(i+1)}]},x\ge 0,$$

where $I_A\left(x\right)=1$ if $x\in A.$ Upon recalling (8), the empirical FGCRE estimator for the consecutive k-out-of-n:G system can be obtained as

$$\begin{array}{ccc}\hfill {\widehat{\mathcal{E}}}_{\alpha ,2}\left(T_{k|n:G}\right)& =& {\int }_0^{\infty }{\psi }_{\alpha }((n-k+1)S_N^k\left(x\right)-(n-k)S_N^{k+1}\left(x\right))dx\hfill \\ & =& \sum _{i=1}^{N-1}{\int }_{X_{i:N}}^{X_{(i+1)}}{\psi }_{\alpha }((n-k+1)S_N^k\left(x\right)-(n-k)S_N^{k+1}\left(x\right))dx\hfill \\ & =& \sum _{i=1}^{N-1}{\psi }_{\alpha }\left((n-k+1){\left({\displaystyle \frac{N-i}{N}}\right)}^k-(n-k){\left({\displaystyle \frac{N-i}{N}}\right)}^{k+1}\right)D_{i+1},\hfill \end{array}$$

(28)

where $D_{i+1}=X_{i+1:N}-X_{i:N}$, $i=1,2,\dots ,N-1,$ denotes the sample spacings.

To evaluate the performance of the proposed estimators ${\widehat{\mathcal{E}}}_{\alpha ,1}\left(T_{k|n:G}\right)$ and ${\widehat{\mathcal{E}}}_{\alpha ,2}\left(T_{k|n:G}\right)$, we conduct a simulation study with exponential data. We assess the average bias and root mean square error (RMSE) for sample sizes of $N=20,30,40,50,100$ and various values of k and $n.$

To assess the performance of the proposed estimators ${\widehat{\mathcal{E}}}_{\alpha ,1}\left(T_{k|n:G}\right)$ and ${\widehat{\mathcal{E}}}_{\alpha ,2}\left(T_{k|n:G}\right),$ we perform a simulation study by employing exponential data. We evaluate the average bias and root mean square error (RMSE) of both estimators for different sample sizes ($N=20,30,40,50,100$), various values of k and n, and $\alpha .$ To specify the value of the smoothing parameter m for a given sample size N, we use the heuristic formula $m=[\sqrt{N}+0.5]$, where $\left[x\right]$ denotes the integer part of x. The simulation involves $5,000$ iterations, and the results are presented in Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7. After analyzing the information presented in the given tables, we have derived the following outcomes:

• For all k and $n,$ as the sample size N increases, both bias and RMSE of the estimators decrease.
• For fixed n and $N,$ as the number of consecutive working components k increases, both bias and RMSE of the estimator increase.

Generally, we can conclude that the number of components n and the number of consecutive working components k affect the efficiency of the estimator.

4.1. Real Data Analysis

We apply the given estimator to real data to examine how closely the FGCRE estimators from consecutive k-out-of-n:G systems align with the theoretical entropy value. The dataset consists of 15 observations of time intervals between successive failures of air conditioning equipment in a Boeing 720 as: 74, 57, 48, 29, 502, 12, 70, 21, 29, 386, 59, 27, 153, 26, 326. This data is modeled using the exponential distribution with the p.d.f

$$f\left(x\right)=\lambda e^{-\lambda x},x>0,\lambda >0.$$

As described by Shanker et al. [28], we analyzed the dataset using the maximum likelihood estimator method to estimate the parameter $\lambda$, resulting in $\widehat{\lambda }=0.00825.$ Additionally, we calculated the Kolmogorov-Smirnov statistic, yielding a value of $0.277$ and a p-value of $0.1662.$ These statistics confirm the goodness-of-fit between the observed data and the fitted exponential distribution.

Table 8 presents various combinations of $k,$n and $\alpha .$ The results indicate that there is a close agreement between the theoretical entropy value and its estimation when the number of functioning components approaches half of the total number of components (n). This observation suggests that the accuracy of the entropy estimation is higher when the system operates with approximately half of its components in a working state.

5. Conclusion

In this study, we explored the utilization of the FGCRE concept within consecutive k-out-of-n:G systems. So, it is highlighted that a strong relationship exists between the FGCRE of such systems derived from continuous and uniform distributions. The obtained expression is so useful for the computation of the FGCRE of the system’s lifetime. However, deriving closed-form expressions for FGCRE becomes intricate when dealing with systems for systems with large component lifetimes or having complicated reliability functions. To address this challenge, we introduced a set of bounds for the FGCRE of consecutive k-out-of-n:G systems. The given bounds serve as essential resources for researchers to analyze FGCRE behaviors effectively. Moreover, we introduced two nonparametric estimators for consecutive k-out-of-n:G systems, specifically applicable to real-world applications. In summary, this research significantly advances the understanding of FGCRE within consecutive k-out-of-n:G systems. The insights gained from this study can be extended to other information metrics, such as fractional generalized cumulative entropy, cumulative entropy, cumulative residual Tsallis entropy, and cumulative Tsallis entropy.