Information Properties of Consecutive Systems using Fractional Generalized Cumulative Residual Entropy

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Information Properties of Consecutive Systems using Fractional Generalized Cumulative Residual Entropy

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Submitted:

12 August 2024

Posted:

14 August 2024

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Abstract

We investigate some information properties of consecutive k-out-of-n:G systems in light of fractional generalized cumulative residual entropy. We firstly derive a formula to compute fractional generalized cumulative residual entropy related to the system's lifetime and explore its preservation properties in terms of established stochastic orders. Additionally, we obtain useful bounds. To aid practical applications, we propose two non parametric estimators for the fractional generalized cumulative residual entropy in these systems. The efficiency and performance of these estimators are illustrated using simulated and real datasets.

Keywords:

Subject: Computer Science and Mathematics - Probability and Statistics

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

2 k \geq 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 (x)

, cumulative distribution function (c.d.f)

F (x)

, and survival (reliability) function

S (x) = P (X > x)

. We denote the system’s lifetime by

T_{k | n : G}

. When

2 k \geq 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} (x) = (n - k + 1) S^{k} (x) - (n - k) S^{k + 1} (x), 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 (X) = E [- log f (X)],

where “log" means for the natural logarithm, with convention

0 log 0 = 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:

E (X) = - \int_{0}^{\infty} S (x) log S (x) d x = \int_{0}^{\infty} S (x) Λ (x) d x,

(2)

where

Λ (x) = - log S (x) = \int_{0}^{x} η (u) d u, x > 0,

(3)

is the cumulative hazard function and

η (u) = f (u) / S (u), 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:

E_{α} (X) = \frac{1}{Γ (α + 1)} \int_{0}^{\infty} S (x) {[- log S (x)]}^{α} d x = \int_{0}^{1} \frac{ψ_{α} (u)}{f (F^{- 1} (u))} d u,

(4)

where

ψ_{α} (x) = \frac{x {(- log x)}^{α}}{Γ (α + 1)}, 0 \leq x \leq 1

(5)

for all

α \geq 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

α

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 (T_{k | n : G})

. It is known that the corresponding transformations of the system’s components, denoted as

U_{i} = F (X_{i})

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

2 k \geq n

, the survival function of

U_{k | n : G}

is given by

{\bar{G}}_{k | n : G} (u) = (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

2 k \geq n,

the FGCRE of

T_{k | n : G},

can be expressed as follows:

E_{α} (T_{k | n : G}) = \int_{0}^{1} \frac{ψ_{α} ({\bar{G}}_{k | n : G} (u))}{f (F^{- 1} (u))} d u,

(7)

where

ψ_{α} (x)

and

{\bar{G}}_{k | n : G} (u)

are defined in (5) and (6), respectively.

Proof.

Applying the transformation

u = F (x)

and referring to (1) and (4), we get

\begin{matrix} E_{α} (T_{k | n : G}) & = & \frac{1}{Γ (α + 1)} \int_{0}^{\infty} S_{k | n : G} (x) {[- log S_{k | n : G} (x)]}^{α} d x \\ = & \int_{0}^{\infty} ψ_{α} (S_{k | n : G} (x)) d x \\ = & \int_{0}^{\infty} ψ_{α} ((n - k + 1) S^{k} (x) - (n - k) S^{k + 1} (x)) d x \\ = & \int_{0}^{1} \frac{ψ_{α} ((n - k + 1) {(1 - u)}^{k} - (n - k) {(1 - u)}^{k + 1})}{f (F^{- 1} (u))} d u \end{matrix}

(8)

\begin{matrix} = & \int_{0}^{1} \frac{ψ_{α} ({\bar{G}}_{k | n : G} (u))}{f (F^{- 1} (u))} d u, \end{matrix}

(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 (x) = \frac{2}{λ} {(1 + \frac{x}{λ})}^{- 3}, x > 0,

(10)

where

λ > 0

denotes the scale parameters. It is clear that

f (F^{- 1} (u)) = \frac{2}{λ} {(1 - u)}^{\frac{3}{2}}

for all

0 < u < 1

. Through algebraic manipulations, we can derive the following expression

\begin{matrix} E_{α} (T_{3 | 5 : G}) = \frac{λ}{2} \int_{0}^{1} ψ_{α} ({\bar{G}}_{k | n : G} (u)) {(1 - u)}^{- \frac{3}{2}} d u . \end{matrix}

(11)

It is clear that the FGCRE is an increasing function of the scale parameter

λ

for all

α \geq 0 .

This means that as the scale parameter

λ

increases, the system’s uncertainty concerning the FGCRE increases. So, this gives the significant impact of the Lomax distribution with the scale parameter

λ

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)

R_{+} = {X; X \geq 0} .

Definition 1.

Let

X, Y \in R_{+}

be r.v.s with pdfs

f_{X} (x)

and

f_{Y} (x),

cdfs

F_{X} (x)

and

F_{Y} (x),

survival functions

S_{X} (x)

and

S_{Y} (x)

and hazard rate orders

λ_{X} (x) = f_{X} (x) / S_{X} (x)

and

λ_{Y} (x) = f_{Y} (x) / S_{Y} (x),

respectively. Then, we say that

1.: X has decreasing failure rate (DFR) if $λ_{X} (x)$ is decreasing in $x;$
2.: X is smaller than Y in the hazard rate order (denoted by $X \leq_{h r} Y$ ) if $λ_{X} (x) \geq λ_{Y} (x)$ for all $x > 0;$
3.: X is smaller than Y in the dispersive order (denoted by $X \leq_{d} Y$ ) if $F_{X}^{- 1} (v) - F_{X}^{- 1} (u) \leq F_{Y}^{- 1} (v) - F_{Y}^{- 1} (u), 0 < u \leq v < 1;$
4.: X is smaller than Y in the location-independent riskier order (denoted by $X \leq_{l i r} Y$ ) if $\int_{0}^{F_{X}^{- 1} (p)} F_{X} (x) d x \leq \int_{0}^{F_{Y}^{- 1} (p)} F_{Y} (x) d x, p \in (0, 1);$

The dispersive

\leq_{d}

order was initially explored by Bickel and Lehmann [22] in nonparametric statistics, while the

\leq_{l i r}

order was proposed by Jewitt [23] in expected utility theory and its applications in insurance. We recall that

X \leq_{d} Y

is equivalent to

f_{Y} (F_{Y}^{- 1} (v)) \leq f_{X} (F_{X}^{- 1} (v)), 0 < v < 1 .

(12)

Additionally, the following implications are well-recognized:

\begin{matrix} if X \leq_{h r} Y and either X or Y is DFR & ⟹ & X \leq_{d} Y ⟹ X \leq_{l i r} Y . \end{matrix}

(13)

Since

ψ_{α} (x) \geq 0

for all

0 \leq x \leq 1

and

α \geq 0

, relations (4) and (12) imply that

E_{α} (X) \leq E_{α} (Y)

when

X \leq_{d} Y

. Consequently, from implication (13), we obtain the following corollary.

Corollary 1.

X \leq_{h r} Y

and either X or Y is DFR, then

E_{α} (X) \leq E_{α} (Y) .

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

η_{Z} (z) = \int_{0}^{z} H (v) d v, z > 0 .

(14)

Landsberger and Meilijson [24] showed that

X \leq_{l i r} Y ⟺ η_{Y}^{- 1} (z) - η_{X}^{- 1} (z) is increasing 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

2 k \geq 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} (x)

and c.d.f

F_{X} (x) .

If X is DFR, then

E_{α} (X_{1 : k}) \leq E_{α} (T_{k | n : G})

for all

α \geq 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} \leq_{h r} 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

2 k \geq 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} (x)

and

f_{Y} (x)

and cdfs

F_{X} (x)

and

F_{Y} (x),

respectively. If

X \leq_{d} Y,

then

E_{α} (T_{k | n : G}^{X}) \leq E_{α} (T_{k | n : G}^{Y})

for all

α \geq 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 (x) = 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} (x) = e^{- x}, x > 0

. It is clear that

λ_{X} (x) = 1 + a (1 - e^{- x})

and

λ_{Y} (x) = 1

. Comparing the hazard rate functions shows that

λ_{X} (x) > λ_{Y} (x)

for

a > 0

, indicating that

X \leq_{h r} Y

. Given that Y has the DFR property, relation (13) leads to

X \leq_{d} Y

. Consequently, by Corollary 3, we have

E_{α} (T_{k | n : G}^{X}) \leq E_{α} (T_{k | n : G}^{Y})

for all

α \geq 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

ϕ_{k | n} (t) = (n - k + 1) {(1 - t)}^{k} - (n - k) {(1 - t)}^{k + 1}, 0 < t < 1 .

For

2 k \geq n,

X \leq_{l i r} Y

and

\frac{ψ_{α} (ϕ_{k | n} (t))}{t}, 0 \leq t \leq 1,

(16)

is a decreasing function of t, then

E_{α} (T_{k | n : G}^{X}) \leq E_{α} (T_{k | n : G}^{Y})

for all

α \geq 0

Proof.

It is clear that Eq. (1) can be represented as

S_{k | n : G} (x) = ϕ_{k | n} (F (x)) .

From (15), we derive

\frac{d}{d z} (η_{Y}^{- 1} (z) - η_{X}^{- 1} (z)) = \frac{1}{F_{Y} (η_{Y}^{- 1} (z))} - \frac{1}{F_{X} (η_{X}^{- 1} (z))} \geq 0,

which implies

F_{X} (z) \geq F_{Y} (η_{Y}^{- 1} (η_{X} (z))),

(17)

for all

z > 0 .

From relations (1) and (4), we obtain

\begin{matrix} \int_{0}^{\infty} \frac{S_{k | n : G}^{X} (x) {[- log S_{k | n : G}^{X} (x)]}^{α}}{Γ (α + 1)} d x & = & \int_{0}^{\infty} ψ_{α} (S_{k | n : G}^{X} (x)) d x \\ = & \int_{0}^{\infty} \frac{ψ_{α} (S_{k | n : G}^{X} (x))}{F_{X} (x)} F_{X} (x) d x \\ = & \int_{0}^{\infty} \frac{ψ_{α} (ϕ_{k | n} (F_{X} (x)))}{F_{X} (x)} F_{X} (x) d x \\ \leq & \int_{0}^{\infty} \frac{ψ_{α} (ϕ_{k | n} (F_{Y} (η_{Y}^{- 1} (η_{X} (x)))))}{F_{Y} (η_{Y}^{- 1} (η_{X} (x)))} F_{X} (x) d x . \end{matrix}

(18)

The inequality arises from Eq. (17) and using the fact that

ψ_{α} (ϕ_{k | n} (t)) / t

is a decreasing function of

0 \leq t \leq 1 .

Furthermore, by setting

u = η_{Y}^{- 1} (η_{X} (x))

, we have

d x = \frac{F_{Y} (u)}{F_{X} (η_{X}^{- 1} (η_{Y} (u)))} d u .

Upon using this, (18) reduces to

\begin{matrix} \int_{η_{Y}^{- 1} (η_{X} (0))}^{\infty} \frac{ψ_{α} (ϕ_{k | n} (F_{Y} (u)))}{F_{Y} (u)} \frac{F_{X} (η_{X}^{- 1} (η_{Y} (u))) F_{Y} (u)}{F_{X} (η_{X}^{- 1} (η_{Y} (u)))} d u \\ = & \int_{η_{Y}^{- 1} (η_{X} (0))}^{\infty} ψ_{α} (ϕ_{k | n} (F_{Y} (u))) d u \\ = & \int_{0}^{\infty} ψ_{α} (S_{k | n : G}^{Y} (x)) d x \\ = & \int_{0}^{\infty} \frac{S_{k | n : G}^{Y} (x) {[- log S_{k | n : G}^{Y} (x)]}^{α}}{Γ (α + 1)} d x . \end{matrix}

The final equality in the previous relation follows from the fact that

η_{Y}^{- 1} (η_{X} (0)) = 0

, implying that

E_{α} (T_{k | n : G}^{X}) \leq E_{α} (T_{k | n : G}^{Y}) .

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

2 k \geq n,

the FGCRE of

T_{k | n : G}

are bounded as follows:

\begin{matrix} B_{α} E_{α} (X_{1}) \leq E_{α} (T_{k | n : G}) \leq D_{α} E_{α} (X_{1}), \end{matrix}

where

B_{α} = {inf}_{u \in (0, 1)} \frac{ψ_{α} ({\bar{G}}_{k | n : G} (u))}{ψ_{α} (u)}

D_{α} = {sup}_{u \in (0, 1)} \frac{ψ_{α} ({\bar{G}}_{k | n : G} (u))}{ψ_{α} (u)} .

Proof.

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

\begin{matrix} E_{α} (T_{k | n : G}) & = \int_{0}^{1} \frac{ψ_{α} ({\bar{G}}_{k | n : G} (u))}{f (F^{- 1} (u))} d u = \int_{0}^{1} \frac{ψ_{α} ({\bar{G}}_{k | n : G} (u))}{ψ_{α} (u)} \frac{ψ_{α} (u)}{f (F^{- 1} (u))} d u \\ \leq sup_{u \in (0, 1)} \frac{ψ_{α} ({\bar{G}}_{k | n : G} (u))}{ψ_{α} (u)} \int_{0}^{1} \frac{ψ_{α} (u)}{f (F^{- 1} (u))} d u = D_{α} E_{α} (X_{1}) . \end{matrix}

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

ψ_{α} (u) .

Theorem 6.

Let

T_{k | n : G}

be the lifetime of consecutive k-out-of-n:G system having the common pdfs

f_{X} (x)

and c.d.f

F_{X} (x) .

If S is the support of f,

m = {inf}_{x \in S} f (x)

and

M = {sup}_{x \in S} f (x)

, then

\begin{matrix} \frac{E_{α} (U_{k | n : G})}{M} \leq E_{α} (T_{k | n : G}) \leq \frac{E_{α} (U_{k | n : G})}{m}, \end{matrix}

(19)

where

E_{α} (U_{k | n : G}) = \int_{0}^{1} ψ_{α} ({\bar{G}}_{k | n : G} (u)) d u .

Proof.

Since

m \leq f (F^{- 1} (u)) \leq M, 0 < u < 1

, we have

\begin{matrix} \frac{1}{M} \int_{0}^{1} ψ_{α} ({\bar{G}}_{k | n : G} (u)) d u \leq \int_{0}^{1} \frac{ψ_{α} ({\bar{G}}_{k | n : G} (u))}{f (F^{- 1} (u))} d u \leq \frac{1}{m} \int_{0}^{1} ψ_{α} ({\bar{G}}_{k | n : G} (u)) d u . \end{matrix}

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

The term

E_{α} (U_{k | n : G})

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 \leq j < r \leq 10 .

Assuming an exponential distribution with mean

μ

, it is easy to see that

m = 0

and

M = 1 / μ .

Moreover, one can see that

B_{α} = 0

for all

α \geq 0 .

On the other hand, one can obtain

E_{α} (X_{1}) = μ

for all

α \geq 0 .

Through algebraic manipulations, we can derive the following expression

\begin{matrix} E_{α} (T_{6 | 10 : G}) = μ \int_{0}^{1} \frac{ψ_{α} ({\bar{G}}_{k | n : G} (u))}{(1 - u)} d u, \end{matrix}

(20)

for all

α \geq 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} ζ (x) d x = 0

for all

n \geq 0,

then

ζ (x) = 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

{k_{j}}_{j \geq 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} (x)

and

f_{Y} (x)

and cdfs

F_{X} (x)

and

F_{Y} (x),

respectively. Then

F_{X}

and

F_{Y}

are members of one family of distributions if, and only if, for a fixed

n,

E_{α} (T_{k | n : G}^{X}) = E_{α} (T_{k | n : G}^{Y}),

(21)

for all

2 k \geq 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{matrix} E_{α} (T_{k | n : G}^{X}) & = & \int_{0}^{1} \frac{ψ_{α} ({\bar{G}}_{k | n : G} (u))}{f_{X} (F_{X}^{- 1} (u))} d u \\ = & - \int_{0}^{1} {(1 - u)}^{2 k - n} \frac{{(1 - u)}^{n - k} ((n - k + 1) - (n - k) (1 - u)) log ({\bar{G}}_{k | n : G} (u))}{f_{X} (F_{X}^{- 1} (u))} d u \\ = & \int_{0}^{1} z^{2 k - n} \frac{Ω (z)}{f_{X} (F_{X}^{- 1} (1 - z))} d z, (taking z = 1 - u) \end{matrix}

(22)

where

Ω (z) = - z^{n - k} ((n - k + 1) - (n - k) z) log ({\bar{G}}_{k | n : G} (z)), 0 < z < 1 .

Similar argument holds for

E_{α} (T_{k | n : G}^{Y}) .

From (22), we have

\int_{0}^{1} [\frac{1}{f_{X} (F_{X}^{- 1} (1 - z))} - \frac{1}{f_{Y} (F_{Y}^{- 1} (1 - z))}] z^{k} Ω (z) d z = 0 .

According to Lemma 1, we can conclude that

f_{X} (F_{X}^{- 1} (1 - z)) = f_{Y} (F_{Y}^{- 1} (1 - z)), a . e . z \in (0, 1),

or equivalently

f_{X} (F_{X}^{- 1} (v)) = f_{Y} (F_{Y}^{- 1} (v)), 0 < v < 1 .

So, it follows that

F_{X}^{- 1} (v) = F_{Y}^{- 1} (v) + d

where d is a constant. By noting that

{lim}_{v \to 0} F_{X}^{- 1} (v) = {lim}_{v \to 0} F_{Y}^{- 1} (v) = 0

for all

v \in (0, 1),

we have

F_{X}^{- 1} (v) = F_{Y}^{- 1} (v) .

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,

E_{α} (T_{n | n : G}^{X}) = E_{α} (T_{n | n : G}^{Y}),

for all

n \geq 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,

\frac{E_{α} (T_{k | n : G}^{X})}{E_{α} (X)} = \frac{E_{α} (T_{k | n : G}^{Y})}{E_{α} (Y)},

(23)

for all

2 k \geq n .

Proof.

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

\frac{E_{α} (T_{k | n : G}^{X})}{E_{α} (X)} = \int_{0}^{1} z^{2 k - n} \frac{Ω (z)}{E_{α} (X) f_{X} (F_{X}^{- 1} (1 - z))} d z .

(24)

Similar argument can be obtained for

E_{α} (T_{k | n : G}^{Y}) / E_{α} (Y) .

From (23) and (24), we obtain

\int_{0}^{1} z^{2 k - n} \frac{Ω (z)}{E_{α} (X) f_{X} (F_{X}^{- 1} (1 - z))} d z = \int_{0}^{1} z^{2 k - n} \frac{Ω (z)}{E_{α} (Y) f_{Y} (F_{Y}^{- 1} (1 - z))} d z .

(25)

Let us set

c = E_{α} (Y) / E_{α} (X) .

Then, (25) can be expressed as

\int_{0}^{1} [\frac{1}{f_{X} (F_{X}^{- 1} (1 - z))} - \frac{1}{c f_{Y} (F_{Y}^{- 1} (1 - z))}] z^{2 k - n} Ω (z) d z = 0 .

Applying Lemma 1, we can conclude that

f_{X} (F_{X}^{- 1} (1 - z)) = c f_{Y} (F_{Y}^{- 1} (1 - z)), a . e . z \in (0, 1),

or equivalently

f_{X} (F_{X}^{- 1} (v)) = c f_{Y} (F_{Y}^{- 1} (v)), 0 < v < 1 .

Given the preceding discussion, it follows that

F_{X}^{- 1} (v) = c F_{Y}^{- 1} (v) + d .

By noting that

{lim}_{v \to 0} F_{X}^{- 1} (v) = {lim}_{v \to 0} F_{Y}^{- 1} (v) = 0

for all

v \in (0, 1),

we have

F_{X}^{- 1} (v) = c F_{Y}^{- 1} (v) .

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,

\frac{E_{α} (T_{n | n : G}^{X})}{E_{α} (X)} = \frac{E_{α} (T_{n | n : G}^{Y})}{E_{α} (Y)},

for all

n \geq 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} \leq X_{2 : N} \leq \dots \leq X_{N : N}

denote their order statistics. By recalling (7), the FGCRE of

T_{k | n : G}

can be rewritten as

\begin{matrix} E_{α} (T_{k | n : G}) & = & \int_{0}^{1} \frac{ψ_{α} ({\bar{G}}_{k | n : G} (u))}{f (F^{- 1} (u))} d u = \int_{0}^{1} ψ_{α} ({\bar{G}}_{k | n : G} (u)) [\frac{d F^{- 1} (u)}{d u}] d u \\ = & \int_{0}^{1} ψ_{α} ((n - k + 1) {(1 - u)}^{k} - (n - k) {(1 - u)}^{k + 1}) [\frac{d F^{- 1} (u)}{d u}] d u, \end{matrix}

(26)

which holds for

2 k \geq n .

Based on (26), we estimate the FGCRE

E_{α} (T_{k | n : G}) .

This approach uses derivative estimates of the inverse distribution function at the sample points. Following Vasicek [27], we estimate the derivative

\frac{d F^{- 1} (u)}{d u}

by approximating it as the slope, defined as:

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

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 \leq N / 2 .

This allows us to obtain the first estimator for

E_{α} (T_{k | n : G})

given by

\begin{matrix} {\hat{E}}_{α, 1} (T_{k | n : G}) & = & \frac{1}{N} \sum_{i = 1}^{N} ψ_{α} ({\bar{G}}_{k | n : G} (\frac{i}{N + 1})) (\frac{N (X_{i + m : N} - X_{i - m : N})}{2 m}) \\ = & \frac{1}{N} \sum_{i = 1}^{N} ψ_{α} ((n - k + 1) {(1 - \frac{i}{N + 1})}^{k} - (n - k) {(1 - \frac{i}{N + 1})}^{k + 1}) \\ \times & (\frac{N (X_{i + m : N} - X_{i - m : N})}{2 m}), \end{matrix}

(27)

for all

2 k \geq n

and

α \geq 0 .

We now introduce the second estimator based on the empirical survival function for the sample corresponding to

S (x)

S_{N} (x) = \sum_{i = 1}^{N - 1} \frac{N - i}{N} I_{[X_{i : N}, x_{(i + 1)}]}, x \geq 0,

where

I_{A} (x) = 1

x \in A .

Upon recalling (8), the empirical FGCRE estimator for the consecutive k-out-of-n:G system can be obtained as

\begin{matrix} {\hat{E}}_{α, 2} (T_{k | n : G}) & = & \int_{0}^{\infty} ψ_{α} ((n - k + 1) S_{N}^{k} (x) - (n - k) S_{N}^{k + 1} (x)) d x \\ = & \sum_{i = 1}^{N - 1} \int_{X_{i : N}}^{X_{(i + 1)}} ψ_{α} ((n - k + 1) S_{N}^{k} (x) - (n - k) S_{N}^{k + 1} (x)) d x \\ = & \sum_{i = 1}^{N - 1} ψ_{α} ((n - k + 1) {(\frac{N - i}{N})}^{k} - (n - k) {(\frac{N - i}{N})}^{k + 1}) D_{i + 1}, \end{matrix}

(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

{\hat{E}}_{α, 1} (T_{k | n : G})

and

{\hat{E}}_{α, 2} (T_{k | n : G})

, 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

{\hat{E}}_{α, 1} (T_{k | n : G})

and

{\hat{E}}_{α, 2} (T_{k | n : G}),

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

α .

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

[x]

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 (x) = λ e^{- λ x}, x > 0, λ > 0 .

As described by Shanker et al. [28], we analyzed the dataset using the maximum likelihood estimator method to estimate the parameter

λ

, resulting in

\hat{λ} = 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

α .

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.

Acknowledgments

This work was supported by Researchers Supporting Project number (RSP2024R464), King Saud University, Riyadh, Saudi Arabia.

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Table 1. The exact value and bounds for

E_{α} (T_{6 | 10 : G})

for different choices of

α .

Table 1. The exact value and bounds for

E_{α} (T_{6 | 10 : G})

for different choices of

α .

$α$	$E_{α} (T_{6 \| 10 : G})$	$D_{α} E_{α} (X_{1})$	$E_{α} (U_{6 \| 10 : G}) / M$
0.1	0.253790 $μ$	4937.445 $μ$	0.203247 $μ$
0.2	0.246750 $μ$	1611.792 $μ$	0.193446 $μ$
0.5	0.230447 $μ$	56.069 $μ$	0.169647 $μ$
0.8	0.219056 $μ$	1.440 $μ$	0.151681 $μ$
1.0	0.213247 $μ$	1.101 $μ$	0.141883 $μ$
1.5	0.202674 $μ$	1.076 $μ$	0.122404 $μ$
2.0	0.195588 $μ$	1.827 $μ$	0.107592 $μ$
2.5	0.190529 $μ$	4.749 $μ$	0.095717 $μ$
3.0	0.186745 $μ$	18.264 $μ$	0.085853 $μ$

Table 2. The Bias and RMSE of the first estimator

{\hat{E}}_{α, 1} (T_{k | n : G}) .

Table 2. The Bias and RMSE of the first estimator

{\hat{E}}_{α, 1} (T_{k | n : G}) .

$α = 0.1$		$N = 30$		$N = 40$		$N = 50$		$N = 100$
n	k	Bias	RMSE	Bias	RMSE	Bias	RMSE	Bias	RMSE
5	3	-0.010898	0.105216	-0.009579	0.089654	-0.004854	0.080614	-0.004425	0.057484
	4	-0.012101	0.073296	-0.008686	0.063602	-0.007962	0.056373	-0.002357	0.039808
	5	-0.011342	0.056113	-0.007421	0.048443	-0.007876	0.044139	-0.003220	0.031291
6	3	-0.009440	0.117519	-0.004641	0.098258	-0.004978	0.087909	-0.003282	0.061718
	4	-0.010196	0.081712	-0.008803	0.070134	-0.006333	0.063083	-0.002612	0.045212
	5	-0.011518	0.061810	-0.009172	0.054063	-0.007364	0.048630	-0.003255	0.035013
	6	-0.012929	0.051138	-0.009353	0.044888	-0.006708	0.039548	-0.003365	0.028388
7	4	-0.010966	0.090091	-0.008536	0.077953	-0.005575	0.069248	-0.001896	0.049748
	5	-0.011133	0.069115	-0.008268	0.058999	-0.006836	0.053299	-0.002281	0.037963
	6	-0.012696	0.056204	-0.009115	0.048011	-0.006197	0.043047	-0.003890	0.030458
	7	-0.012264	0.045635	-0.008862	0.041127	-0.006831	0.036810	-0.003554	0.026578
8	4	-0.007320	0.095422	-0.005076	0.081741	-0.005717	0.076165	-0.000826	0.053521
	5	-0.011195	0.075112	-0.008280	0.064280	-0.007420	0.057273	-0.003794	0.040709
	6	-0.011476	0.059650	-0.009077	0.051937	-0.006067	0.046792	-0.004184	0.033449
	7	-0.013114	0.050062	-0.008815	0.043506	-0.006897	0.039713	-0.003551	0.028184
	8	-0.011777	0.042585	-0.009081	0.038260	-0.007125	0.033553	-0.003560	0.024383

Table 3. The Bias and RMSE of the first estimator

{\hat{E}}_{α, 1} (T_{k | n : G}) .

Table 3. The Bias and RMSE of the first estimator

{\hat{E}}_{α, 1} (T_{k | n : G}) .

$α = 1$		$N = 30$		$N = 40$		$N = 50$		$N = 100$
n	k	Bias	RMSE	Bias	RMSE	Bias	RMSE	Bias	RMSE
5	3	0.000087	0.087601	0.000872	0.074942	-0.000812	0.066222	-0.000199	0.047229
	4	0.000677	0.061861	0.000951	0.052752	-0.000292	0.048263	-0.000173	0.033817
	5	-0.001023	0.047448	-0.000314	0.041763	0.000521	0.036230	0.000228	0.026700
6	3	0.001409	0.089266	0.000389	0.078280	0.000591	0.069693	0.000525	0.048964
	4	0.000284	0.066270	-0.000330	0.056581	0.000154	0.050492	0.000406	0.036359
	5	0.000046	0.052805	-0.000540	0.045278	0.000055	0.039331	0.000399	0.028064
	6	-0.000570	0.042544	-0.000926	0.037313	-0.000428	0.032658	0.000080	0.023334
7	4	0.000300	0.068752	0.000634	0.060594	0.000478	0.053156	0.000868	0.037872
	5	0.000316	0.055068	-0.000188	0.047579	-0.000385	0.043313	-0.000690	0.030024
	6	-0.001600	0.044820	0.000122	0.038575	-0.000157	0.035549	0.000075	0.024823
	7	-0.000246	0.038482	-0.000109	0.033277	-0.000450	0.029768	-0.000460	0.020956
8	4	0.001810	0.071259	-0.000350	0.061727	-0.001011	0.055154	0.000785	0.039367
	5	-0.000294	0.058607	0.000110	0.049190	-0.000007	0.044643	-0.000307	0.031387
	6	-0.000963	0.048215	-0.000025	0.041604	-0.000243	0.037351	0.000320	0.026658
	7	-0.000449	0.041519	-0.000056	0.034707	-0.000747	0.032291	-0.000337	0.022519
	8	-0.001574	0.035239	-0.000403	0.030317	-0.000894	0.027829	0.000223	0.019494

Table 4. The Bias and RMSE of the first estimator

{\hat{E}}_{α, 1} (T_{k | n : G}) .

Table 4. The Bias and RMSE of the first estimator

{\hat{E}}_{α, 1} (T_{k | n : G}) .

$α = 2$		$N = 30$		$N = 40$		$N = 50$		$N = 100$
n	k	Bias	RMSE	Bias	RMSE	Bias	RMSE	Bias	RMSE
5	3	-0.001090	0.086459	0.000815	0.075403	-0.001139	0.066924	-0.000073	0.047757
	4	0.034278	0.070438	0.026845	0.060862	0.024803	0.052971	0.014174	0.035674
	5	0.014319	0.047563	0.010515	0.040428	0.009683	0.035569	0.005089	0.025548
6	3	0.063920	0.126898	0.062550	0.111951	0.060926	0.102074	0.044846	0.071823
	4	0.038461	0.075607	0.033681	0.066047	0.029243	0.057837	0.015990	0.039400
	5	0.018664	0.052510	0.014582	0.044903	0.012534	0.039484	0.005635	0.027945
	6	0.007147	0.039141	0.005249	0.034102	0.004860	0.029907	0.002433	0.022023
7	4	0.041086	0.083484	0.036617	0.070899	0.032951	0.062740	0.018144	0.040586
	5	0.021959	0.055792	0.018673	0.048884	0.014572	0.042988	0.008572	0.028540
	6	0.010251	0.042669	0.008163	0.035800	0.006582	0.032881	0.003422	0.023119
	7	0.003633	0.033836	0.003067	0.029107	0.002643	0.026160	0.002013	0.019432
8	4	0.045197	0.085555	0.038435	0.074035	0.037415	0.065820	0.020444	0.042791
	5	0.024856	0.059893	0.019444	0.050514	0.017341	0.044100	0.008817	0.030109
	6	0.012141	0.044384	0.009642	0.038875	0.008753	0.035025	0.003794	0.024345
	7	0.006040	0.036246	0.002859	0.032057	0.003239	0.028222	0.001896	0.020497
	8	0.001058	0.029722	0.001472	0.026524	0.001431	0.024069	0.000941	0.017894

Table 5. The Bias and RMSE of the second estimator

{\hat{E}}_{α, 2} (T_{k | n : G}) .

Table 5. The Bias and RMSE of the second estimator

{\hat{E}}_{α, 2} (T_{k | n : G}) .

$α = 0.1$		$N = 30$		$N = 40$		$N = 50$		$N = 100$
n	k	Bias	RMSE	Bias	RMSE	Bias	RMSE	Bias	RMSE
5	3	-0.010609	0.102546	-0.008270	0.088836	-0.005443	0.080873	-0.002863	0.058167
	4	-0.010304	0.073576	-0.010357	0.063081	-0.006064	0.057594	-0.003082	0.041216
	5	-0.014063	0.057050	-0.009564	0.049681	-0.007636	0.043591	-0.003881	0.031672
6	3	-0.007927	0.114087	-0.005274	0.099831	-0.002167	0.088074	-0.002003	0.062661
	4	-0.010957	0.081463	-0.007744	0.071106	-0.006705	0.065384	-0.003142	0.044996
	5	-0.012472	0.061216	-0.008003	0.054332	-0.006736	0.048667	-0.002746	0.034079
	6	-0.013312	0.049958	-0.008733	0.044566	-0.006949	0.040256	-0.003810	0.028654
7	4	-0.010511	0.090065	-0.008348	0.078439	-0.008723	0.068968	-0.002051	0.048977
	5	-0.012018	0.068814	-0.008633	0.059495	-0.004883	0.053448	-0.004237	0.038646
	6	-0.011984	0.054471	-0.007330	0.048398	-0.006811	0.043351	-0.002883	0.031013
	7	-0.010979	0.045610	-0.009159	0.039701	-0.007405	0.036903	-0.003332	0.026529
8	4	-0.008034	0.097000	-0.004299	0.083309	-0.006195	0.075089	-0.001807	0.052764
	5	-0.010637	0.075050	-0.008024	0.065490	-0.007708	0.058203	-0.003380	0.041227
	6	-0.011068	0.060779	-0.007946	0.052276	-0.006342	0.047288	-0.003544	0.033142
	7	-0.011923	0.049250	-0.007979	0.043511	-0.006613	0.038337	-0.003617	0.027815
	8	-0.012459	0.043427	-0.008960	0.037006	-0.007978	0.034281	-0.003476	0.023718

Table 6. The Bias and RMSE of the second estimator

{\hat{E}}_{α, 2} (T_{k | n : G}) .

Table 6. The Bias and RMSE of the second estimator

{\hat{E}}_{α, 2} (T_{k | n : G}) .

$α = 1$		$N = 30$		$N = 40$		$N = 50$		$N = 100$
n	k	Bias	RMSE	Bias	RMSE	Bias	RMSE	Bias	RMSE
5	3	0.000212	0.085861	-0.000383	0.074161	0.000658	0.067124	0.001064	0.046434
	4	0.000873	0.060747	-0.001359	0.053834	-0.001447	0.047587	-0.000068	0.033720
	5	0.000465	0.047697	0.000799	0.041411	-0.000355	0.036525	0.000007	0.026548
6	3	-0.000611	0.090382	-0.000357	0.078913	0.000484	0.069387	0.000170	0.049160
	4	-0.000483	0.067131	0.000767	0.057582	0.000042	0.051330	0.000508	0.035924
	5	-0.000430	0.051884	-0.000213	0.045401	-0.000394	0.040414	-0.000242	0.028758
	6	-0.000387	0.042309	-0.000377	0.037006	-0.000479	0.032480	-0.000493	0.023257
7	4	-0.000453	0.068868	-0.000341	0.060492	-0.000269	0.052825	0.000865	0.037833
	5	0.000392	0.056091	0.000053	0.047779	-0.000847	0.042480	-0.000198	0.030513
	6	-0.000500	0.045534	-0.000440	0.039434	-0.000049	0.035552	-0.000028	0.025303
	7	-0.000535	0.038237	-0.000614	0.033145	-0.000271	0.029834	0.000198	0.021260
8	4	0.001655	0.071870	-0.001471	0.062020	-0.000568	0.056329	0.000196	0.039943
	5	-0.000569	0.056866	-0.000085	0.049564	-0.000235	0.045120	-0.000662	0.031710
	6	-0.000216	0.048526	-0.000433	0.041005	-0.001622	0.037182	-0.000601	0.026537
	7	-0.000753	0.040824	-0.000632	0.035827	-0.000771	0.031846	-0.000160	0.022491
	8	-0.001067	0.035046	-0.000742	0.031062	-0.000528	0.027216	0.000628	0.019584

Table 7. The Bias and RMSE of the second estimator

{\hat{E}}_{α, 2} (T_{k | n : G}) .

Table 7. The Bias and RMSE of the second estimator

{\hat{E}}_{α, 2} (T_{k | n : G}) .

$α = 2$		$N = 30$		$N = 40$		$N = 50$		$N = 100$
n	k	Bias	RMSE	Bias	RMSE	Bias	RMSE	Bias	RMSE
5	3	0.001317	0.086416	-0.001661	0.076015	0.001307	0.067851	-0.000699	0.047553
	4	-0.000150	0.060925	-0.000304	0.051711	-0.000440	0.047292	-0.000288	0.033949
	5	-0.000058	0.045216	0.000112	0.040160	0.000627	0.035967	0.000482	0.025006
6	3	0.000443	0.093072	-0.001661	0.080002	-0.001348	0.071543	-0.000287	0.049726
	4	0.000374	0.064174	0.000945	0.055211	0.000121	0.050066	0.000298	0.035013
	5	0.000484	0.050087	-0.000586	0.042798	0.000282	0.038479	0.000650	0.027301
	6	-0.000109	0.040256	-0.000478	0.034969	0.000093	0.031228	-0.000690	0.022026
7	4	-0.000368	0.067255	-0.000452	0.056703	-0.001297	0.051753	-0.000065	0.037156
	5	-0.001662	0.051118	0.000641	0.045238	0.000413	0.040474	0.000762	0.028545
	6	-0.000668	0.042594	-0.000386	0.036797	0.000147	0.032351	-0.000145	0.023491
	7	0.000545	0.035956	-0.000206	0.030602	-0.000309	0.028139	0.000132	0.019843
8	4	-0.000954	0.068949	0.000744	0.059647	-0.000047	0.052899	0.000328	0.038041
	5	-0.000855	0.053399	0.000319	0.047349	-0.000043	0.041896	0.000557	0.029469
	6	0.000340	0.044608	-0.000014	0.038450	0.000239	0.034654	-0.000300	0.024364
	7	0.000250	0.037999	0.000033	0.032327	0.000410	0.029667	-0.000122	0.020663
	8	0.000277	0.033181	0.000718	0.028420	-0.000165	0.025314	-0.000493	0.018187

Table 8. Comparison of theoretical values and estimates of FGCRE of

T_{k | 6 : G}

based on exponential distribution for successive failures of air conditioning equipment in a Boeing 720.

Table 8. Comparison of theoretical values and estimates of FGCRE of

T_{k | 6 : G}

based on exponential distribution for successive failures of air conditioning equipment in a Boeing 720.

$α$	k	$E_{α} (T_{k \| 5 : G})$	${\hat{E}}_{α, 1} (T_{k \| 5 : G})$	${\hat{E}}_{α, 2} (T_{k \| 5 : G})$	$E_{α} (T_{k \| 6 : G})$	${\hat{E}}_{α, 1} (T_{k \| 6 : G})$	${\hat{E}}_{α, 2} (T_{k \| 6 : G})$
	3	58.669550	44.189875	34.759008	65.757406	53.069440	40.154187
0.1	4	36.091734	19.451760	18.285162	41.569942	24.070849	21.252137
	5	24.242424	9.7693280	11.908400	28.135564	12.383475	13.764174
	6				20.202020	7.084899	9.981658
	3	49.258087	61.173750	44.983267	50.171584	67.051873	50.106795
1.0	4	34.258325	31.454190	20.858847	36.756520	36.672164	23.812712
	5	24.242424	16.107555	11.917905	27.091908	19.911450	13.792459
	6				20.202020	11.297715	9.424714
	3	45.368069	67.425564	61.815013	45.642327	69.309002	66.956682
2.0	4	33.033300	43.046387	29.345655	34.306351	47.652819	33.495064
	5	24.242424	23.884659	14.469465	26.333470	28.707664	17.290249
	6				20.202020	16.436591	10.152834

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Information Properties of Consecutive Systems using Fractional Generalized Cumulative Residual Entropy

Abstract

1. Introduction

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

2.1. Expression and Stochastic Orders

2.2. Some Bounds

3. Characterization Results

4. Nonparametric Estimation

4.1. Real Data Analysis

5. Conclusion

Acknowledgments

References

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