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\begin{document}
\title{ \textbf{  Extropy Analysis of the Inactivity Times of the Failed Components in Coherent Systems with Applications to Estimation and Image Processing}}
\author{%
  Zohreh Pakdaman$^1$\footnote{Corresponding authors.\newline
  E-mail: {\it zpakdaman@hormozgan.ac.ir} (Z.~Pakdaman),
  {\it reza.alizadehn@hormozgan.ac.ir} (R.~Alizadeh Noughabi).} 
  and
  Reza Alizadeh Noughabi$^{1,*}$\\
  $^1$\,{\small {\it Department of Statistics, University of Hormozgan, Hormozgan,  Iran}}\vspace{-0.2cm}\\%
}


%\date{}
\maketitle
\vspace{-0.4cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
This study presents a comprehensive examination of the extropy associated with the inactivity times of failed components in coherent systems. In contrast to conventional reliability measures that emphasize system inactivity or residual lifetime analysis, the proposed framework captures the underlying informational characteristics embedded in the inactivity times of failed components while the overall system remains operational. By employing system signatures, we derive closed-form representations for the extropy of these inactivity times and establish a series of stochastic bounds and comparative results that elucidate the influence of system structure and configuration.
Furthermore, we assess the sensitivity of extropy through numerical experiments based on Weibull-distributed lifetimes. On the applied side, we develop two non-parametric estimators for the extropy of inactivity times of failed components in coherent systems. The proposed estimators’ efficiency  are demonstrated via simulated datasets and further illustrated through an image processing application, highlighting the practical relevance of the extropy-based approach in reliability and information-theoretic analysis.
\end{abstract}

\vskip 1mm \noindent{\bf Keywords and Phrases}: Extropy, Non-parametric estimators, 
   Order statistics, 
 Stochastic order, System signature, Inactivity times of the failed components, Image processing.\\
\noindent{\bf AMS 2000 Subject Classification}: Primary 62N05;
Secondary 62F10.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
In reliability theory, one of the most intricate challenges arises when examining the behavior of components that have already failed but do not immediately lead to the breakdown of the overall system. Classical reliability analyses tend to focus on system lifetime distributions or the survival probabilities of still-functioning components, often neglecting the hidden impact of those elements that have failed yet remain embedded within an operational structure. By emphasizing the concept of inactivity time, the duration during which a component remains failed while the system continues to perform, we gain valuable insights into the system’s intrinsic resilience, interdependence among components, and long-term susceptibility to deterioration. The practical significance of this viewpoint can be illustrated through several distinct scenarios:

1- In a patient monitoring setup consisting of multiple sensors, such as heart rate, temperature, and oxygen level sensors, the system can continue to function even when one or more sensors have stopped working. These failed sensors accumulate an inactivity time while the system keeps monitoring through the remaining active units. Analyzing such durations helps biomedical engineers identify early degradation patterns, predict component lifespans, and design optimal replacement or calibration schedules before the system’s overall reliability is compromised.

2- In large-scale distributed software systems, such as cloud platforms with numerous interdependent service modules, some components (e.g., authentication servers or data caches) may crash without immediately disrupting the entire service. These failed modules remain inactive within the architecture while other modules continue handling requests. Investigating the inactivity times of such modules enables software engineers to assess subsystem stability, determine optimal reboot or patch intervals, and enhance fault-tolerant designs for sustained reliability under load.

3- In urban traffic control systems composed of multiple sensors and controllers at intersections, the network remains operational even if certain sensors fail temporarily. These inactive sensors accumulate inactivity periods before the network’s performance begins to degrade noticeably. Studying their inactivity times helps city engineers detect critical nodes, implement preventive maintenance, and strengthen the system’s robustness against progressive failures.

Together, these examples demonstrate that the concept of inactivity time represents a crucial and complementary reliability measure. It quantifies how long failed components persist within an otherwise functional system, providing both theoretical understanding and practical guidance for enhancing structural robustness, improving maintenance strategies, and preventing cascading failures across complex engineered systems. 

%In many engineering systems, the failure of the overall structure does not necessarily imply the simultaneous failure of all components. Within the framework of coherent systems, it is possible that at the system failure time $t$, certain components remain alive with non-zero residual lifetimes. Investigating these surviving elements, particularly through the lens of cumulative residual extropy (CREX), offers a novel perspective on system behavior and risk management. 
 
For a coherent system consisting of $n$ components with overall lifetime denoted by $T(n)$, 
let the individual component lifetimes $Y_i$ $(i=1,\ldots,n)$ be independent and identically distributed (IID) random variables characterized by a common cumulative distribution function (CDF) $G(\cdot)$. 
The reliability structure of the system can be completely specified by its signature vector, 
an $n$-dimensional probability vector expressed as $\boldsymbol{s} = (s_1, s_2, \ldots, s_n)$, where each element 
$s_i = P\!\left(T(n) = Y_{i:n}\right)$ represents the probability that system failure occurs precisely at the time of the $i$th order statistic $Y_{i:n}$, corresponding to the $i$th weakest component. 
By definition, the components of the signature satisfy the normalization constraint $\sum_{i=1}^{n}s_i = 1$. 
Hence, the signature provides a probabilistic characterization of how the overall system failure is influenced by the order in which individual component failures occur.  
A comprehensive exposition of the system signature framework can be found in Samaniego (2007).

Let the random variable $T(n)$ possess the probability density function (PDF) $g_{T(n)}(\cdot)$ 
and CDF $G_{T(n)}(\cdot)$. 
In this study, we focus on a particular class of coherent systems whose signature vectors take a degenerate form given by 
$
\boldsymbol{s} = (0, \ldots, 0, s_i, s_{i+1}, \ldots, s_n), \quad i = 2, \ldots, n,
$
which corresponds to systems in which failures of the first $(i-1)$ components cannot directly cause system breakdown. 
Throughout the analysis, we assume that the system operation commences at time $t=0$.

% When the system eventually fails at some time $t$, a subset of its components, specifically those with lifetimes $Y_{k:n}$ for $k = i+1, i+2, \ldots, n$, continue to remain operational.
%When analyzing a coherent system that is still operational at time $t$, particular attention is directed toward those components that have already failed before $t$ but are not capable, on their own, of causing the system’s breakdown. For such components, the inactivity time can be represented by the conditional random variable
% $ t-Y_{j:n }|Y_{j:n} < t <T$, where $t > 0$ and $j = 1,\ldots , i$ 
%which quantifies the elapsed duration that a component has remained inactive by time $t$, under the condition that the system continues to function while this component has failed earlier. This formulation offers a rigorous probabilistic framework for describing the hidden aging behavior of failed components that coexist within an active system.
%From a practical standpoint, such a characterization allows engineers to determine how long certain elements persist in an inactive state within a functioning structure, to assess the extent of latent vulnerability introduced by early failures, and to develop informed strategies for proactive maintenance or timely replacement. Inactivity times therefore emerge as a key metric for uncovering concealed system fragilities, optimizing resource allocation, and strengthening long-term reliability planning.

For simplicity
in notation, we define $ t-Y_{j:n }|Y_{j:n} < t <T $, $t-Y_{j:n}| Y_{j:n}< t <Y_{k:n} $, and $ t-Y_{j:n}|Y_{l:n} < t < Y_{l+1:n}$
for $1 \leq j \leq l < k \leq n$ as $IT_{j,n}(t)$, $IY_{j,k,n}(t)$, and $IY^t_{l-j+1:l}$, respectively, throughout the rest of the
paper. Additionally, assume that the CDF of $IT_{j,n}(t)$ (or $IY_{j,k,n}(t)$ or $IY^t_{l-j+1:l}$) is $G_{IT_{j,n}(t)}(.)$ (or $G_{IY_{j,k,n}(t)}(.)$ or $G_{IY^t_{l-j+1:l}}(.)$), respectively. From Goliforushani et al. (2012), we have
\begin{equation}\label{10}
\bar{G}_{IT_{j,n}(t)}(y) = \sum\limits_{k=i}^{n} q_k(t) \bar{G}_{IY_{j,k,n}(t)}(y),
\end{equation}
where
\begin{equation}\label{11}
\bar{G}_{IY_{j,k,n}(t)}(y) = \sum\limits_{l=j}^{k-1} B_{l,j,k,n}(t)\,\bar{G}_{IY^t_{l-j+1:l}}(y),
\end{equation}
\begin{eqnarray}\label{12}
\bar{G}_{IY^t_{l-j+1:l}}(y) &=& \sum\limits_{m=j}^{l} 
\binom{l}{m}
\left( \frac{{G}(t-y)}{{G}(t)} \right)^{m}
\left( 1 - \frac{{G}(t-y)}{{G}(t)} \right)^{l-m}\nonumber\\
 &=&\int_{0}^{\frac{{G}(t-y)}{{G}(t)}}j\binom{l}{j}u^{j-1}(1-u)^{l-j}~du,~0<y<t,
\end{eqnarray}
\begin{equation}\label{13}
q_k(t) = \frac{ s_k \sum\limits_{l=j}^{k-1}\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l}
{ \sum\limits_{m=i}^{n}\sum\limits_{l=j}^{m-1} s_m \binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l },
\end{equation}
and
\begin{equation}\label{14}
B_{l,j,k,n}(t) = \frac{\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l}
{  \sum\limits_{l=j}^{k-1}\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l}.
\end{equation}
 Additionally, assume that the PDF of $IT_{j,n}(t)$ (or $IY_{j,k,n}(t)$ or $IY^t_{l-j+1:l}$) is $g_{IT_{j,n}(t)}(.)$ (or $g_{IY_{j,k,n}(t)}(.)$ or $g_{IY^t_{l-j+1:l}}(.)$), respectively. Therefore, we have 
 \begin{equation}\label{10pdf}
{g}_{IT_{j,n}(t)}(y) = \sum\limits_{k=i}^{n} q_k(t) {g}_{IY_{j,k,n}(t)}(y),
\end{equation}
where
\begin{equation}\label{11pdf}
{g}_{IY_{j,k,n}(t)}(y) = \sum\limits_{l=j}^{k-1} B_{l,j,k,n}(t) {g}_{IY^t_{l-j+1:l}}(y),
\end{equation}
and 
\begin{equation}\label{12pdf}
{g}_{IY^t_{l-j+1:l}}(y) =j\binom{l}{j}\frac{{g}(t-y)}{{G}(t)}(\frac{{G}(t-y)}{{G}(t)})^{j-1}(1-\frac{{G}(t-y)}{{G}(t)})^{l-j},~0<y<t.
\end{equation}
%\begin{equation}\label{15}
%\xi J\left( IT_{j,n}(t) \right) &=&  - \frac{1}{2}\int_0^t \bar{G}^2_{IT_{j,n}(t)}(y)~dy
%\end{equation}


For coherent systems characterized by a signature vector of the form 
$\boldsymbol{s} = (0, \ldots, 0, s_i, s_{i+1}, \ldots, s_n)$ for $i = 2, \ldots, n$, 
the components that have already failed prior to a given time $t$ continue to exert an implicit effect on the overall system performance, 
even though their individual failure times are not directly observable. 
In such cases, the system can be perceived as a "black box" since, while the system remains operational, 
the precise lifetimes of the failed components are hidden from observation. 
To characterize this hidden reliability behavior, we define the conditional probability function
$
\bar{G}_{IT_{j,n}(t)}(y) = P(t - Y_{j:n} > y \mid Y_{j:n} < t < T),
$
which represents the survival function associated with the inactivity times of the failed components. 
This function quantifies the probability that the inactivity duration of the $j$th component, 
which failed before time $t$, exceeds a specified value $y$ given that the system as a whole is still functioning at time $t$. 
The condition $Y_{j:n} < t$ indicates that the $j$th component has already failed before $t$, 
whereas $t < T$ ensures that the system lifetime has not yet expired. 
Consequently, the conditional probability $\bar{G}_{IT_{j,n}(t)}(y)$ 
captures the stochastic behavior of component inactivity within an operating system, 
serving as a fundamental measure of how long failed components remain dormant 
without immediately compromising overall system reliability.

 
A substantial body of research has addressed the study of inactivity times in coherent systems under a variety of modeling assumptions and reliability structures. 
Notable contributions include the works of Khaledi and Shaked (2007), Li and Zhao (2008), Zhang (2010), Zhang and Yang (2010), Goliforushani and Asadi (2011), Goliforushani et al. (2012), Tavangar (2015), Navarro et al. (2017), Navarro and Calì (2019), Guo et al. (2022), Rao and Naqvi (2024), and Zhang (2025). 
These studies collectively provide a broad theoretical and methodological foundation for analyzing the stochastic behavior of component inactivity and its implications for system reliability assessment.


Classical reliability indices, including mean inactivity time and conventional hazard-type measures, often fail to adequately represent the inherent complexity that emerges when certain components of a coherent system have already failed, yet the system as a whole remains operational. 
Such traditional summaries typically overlook the subtle distributional features and latent dependencies that characterize the inactivity times of failed components still embedded within a functioning structure. 
To address this shortcoming, we adopt an information-theoretic framework and introduce the notion of extropy as a fundamental measure for assessing the structural complexity associated with inactivity times.
 Within the broader context of information theory, one of the foundational constructs for quantifying uncertainty in random phenomena is entropy. 
Introduced by Shannon (1948), entropy provides a principled means of measuring the expected uncertainty in a probabilistic model. 
For a continuous random variable $X$ with probability density function (PDF) $f(x)$, the Shannon entropy is defined as
$
H(X) = -E[\log f(X)],
$
where $\log$ denotes the natural logarithm. 
In contrast, extropy is proposed as the dual concept to entropy. 
While entropy traditionally reflects disorder and unpredictability, extropy has been interpreted in subsequent literature as a measure of structural coherence, vitality, and the inherent tendency of systems toward organization, stability, and potential growth. 
Formally, Lad et al. (2015) defined the extropy of a random variable $X$ with PDF $f(x)$ as
\begin{equation}\label{extropybase}
J(X) = -\frac{1}{2}\int_{-\infty}^{+\infty} f^2(x)\,dx = -\frac{1}{2}\int_{0}^{1} f(F^{-1}(u))\,du,
\end{equation}
where $F^{-1}(\cdot)$ denotes the inverse of the CDF of $X$.

A growing body of literature has explored extropy and its applications in analyzing the informational properties of reliability models. 
For a detailed account of the theoretical and applied aspects of extropy within reliability and information theory, the reader is referred to the works of 
Qiu (2017), 
Qiu and Jia (2018a, 2018b), 
Kayal (2019), 
Jose and Sathar (2019), 
Jahanshahi et al. (2020), 
Pakdaman and Hashempour (2021), 
Chakraborty and Pradhan (2022, 2023a, 2023b, 2024), 
and Pakdaman and Noughabi (2025a, 2025b, 2025c).

      


This study focuses on the analysis of extropy associated with the inactivity times of failed components within a coherent system that remains operational at time $t$, as formalized in Equation \eqref{10}. The structure of the paper is as follows: Section 2 provides a comprehensive formulation of the extropy for inactivity durations of failed components in still-functioning systems, employing the conditional coefficients vector and deriving relevant stochastic comparisons and theoretical bounds. Section 3 introduces a new divergence measure aimed at capturing the structural complexity inherent in the inactivity times of failed elements embedded in active systems. In Section 4,
we propose two non-parametric estimators for estimating
the extropy of inactivity times of failed components within a coherent system and demonstrate their application using 
simulated data. Also, in Section 5, the efficiency and performance of these estimators are demonstrated through 
 an image processing application.
 Finally, Section 6 concludes with a synthesis of the main theoretical contributions and practical implications, emphasizing the role of inactivity times of failed components in shaping the long-term reliability of coherent systems.

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Extropy of  inactivity times of the failed components}
Let $Y_1, Y_2, \ldots, Y_n$ denote the IID lifetimes of $n$ components, each following the CDF $G(\cdot)$. 
Consider a coherent system of order $n$ characterized by the signature vector 
$\boldsymbol{s} = (0, \ldots, 0, s_i, s_{i+1}, \ldots, s_n)$ for $i = 2, \ldots, n$. 
Let $T(n)$ represent the system lifetime, with the PDF $g_{T(n)}(\cdot)$ and CDF $G_{T(n)}(\cdot)$. 
 The primary objective is to evaluate the extropy associated with the inactivity times of failed components, 
that is, to compute the extropy of the conditional random variable 
$t - Y_{j:n} \mid Y_{j:n} < t < T$. 
This quantity reflects the informational structure of component inactivity within an operational system, capturing how uncertainty is distributed across the durations for which failed components remain inactive while the system continues to function. 


In this section, we derive an analytical expression for the extropy of the inactivity times of failed components 
in a coherent system whose reliability configuration is specified by the signature vector 
$\boldsymbol{s} = (0, \ldots, 0, s_i, s_{i+1}, \ldots, s_n)$ for $i = 2, \ldots, n$. 
By employing Equations~\eqref{10pdf} and~\eqref{extropybase}, 
the extropy of $IT_{j,n}(t)$ can be formulated as
\begin{eqnarray}\label{15}
J\left( IT_{j,n}(t) \right) &=&  - \frac{1}{2}\int_0^t {g}^2_{IT_{j,n}(t)}(y)~dy\nonumber\\
 &=&- \frac{1}{2}\int_0^t \Big(\sum\limits_{k=i}^{n} q_k(t) {g}_{IY_{j,k,n}(t)}(y)\Big)^2~dy
 \nonumber\\
 &=&- \frac{1}{2}\int_0^t \Big(\sum\limits_{k=i}^{n}\sum\limits_{l=j}^{k-1} q_k(t) B_{l,j,k,n}(t){g}_{IY^t_{l-j+1:l}}(y)\Big)^2~dy
 \nonumber\\
 &=&- \frac{1}{2}\int_0^t \Big(\sum\limits_{k=i}^{n}\sum\limits_{l=j}^{k-1} q_k(t) B_{l,j,k,n}(t)j\binom{l}{j}\frac{{g}(t-y)}{{G}(t)}(\frac{{G}(t-y)}{{G}(t)})^{j-1}(1-\frac{{G}(t-y)}{{G}(t)})^{l-j}\Big)^2~dy
 \nonumber\\
 &=&- \frac{1}{2}\int_0^1 \Big(\sum\limits_{k=i}^{n}\sum\limits_{l=j}^{k-1} q_k(t) B_{l,j,k,n}(t)j\binom{l}{j}(1-u)^{j-1}u^{l-j}\Big)^2~du.
\end{eqnarray}
In this context, $IY^{t}_{l-j+1:l}$ represents the $(l-j+1)$-th order statistic among $l$ independent and identically distributed random variables having the same distribution as $(t - X \mid X < t)$, whose corresponding CDF is given by $\frac{G(t - x)}{G(t)}$. 
 In the subsequent development, our aim is to reexpress the extropy corresponding to the inactivity times of failed components in a form that is analytically more tractable. 
To achieve this, we begin by simplifying specific constituent terms in Equation~\eqref{15}, which facilitates a more compact and interpretable formulation of the overall expression. 
As an initial step in this refinement process, we focus on the multiplicative interaction between $q_k(t)$ and $B_{l,j,k,n}(t)$, for which a reduced form is obtained as follows:

\begin{equation}\label{17}
q_k(t) B_{l,j,k,n}(t)=\frac{s_k \binom{n}{l} (\frac{G(t)}{\bar{G}(t)})^l}{\sum\limits_{m=i}^{n} s_m \binom{n}{l} (\frac{G(t)}{\bar{G}(t)})^l}.
\end{equation}
By  substitution of Equation \eqref{17} into Equation \eqref{15}, we immediately arrive at the following simplified representation:
\begin{eqnarray}\label{18}
 J\left(IT_{j,n}(t) \right) &=& - \frac{1}{2}\int_0^t\Big(\frac{\bar{G}^{n}(t)}{\sum\limits_{m=i}^{n} s_m \big( {G}_{Y_{j:n}}(t) - {G}_{Y_{m:n}}(t) \big)}\sum\limits_{k=i}^{n}s_k\sum\limits_{l=j}^{k-1} j\binom{l}{j}\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l\nonumber\\
 &\times &(1-u)^{j-1}u^{l-j}~du\Big)^2~du.
\end{eqnarray}
Equation \eqref{18} establishes a systematic framework for computing the extropy of the inactivity times of failed components in a coherent system that continues to operate at time $t$. 
In this framework, the relative contributions of inactive components are quantified through the conditional coefficient vector 
$\mathbf{q}(t) = (0, 0, \dots, 0, q_i(t), \dots, q_n(t))$, 
which serves as the system's conditional signature. 
Each entry $q_k(t)$ represents the probability that the $k$th failed component, with lifetime $Y_{k:n}$, influences the inactivity structure, conditional on the system remaining functional while components $Y_{k:n}$ for $k=1, \dots, i$ have already failed. 
This approach provides reliability engineers with a nuanced understanding of the interdependence between failed and surviving components, 
emphasizing how the inactivity of lower-order elements interacts with the continuing operation of higher-order components. 
Consequently, the conditional signature vector $\mathbf{q}(t)$ offers a valuable analytical tool for evaluating latent system vulnerabilities 
and for designing systems where distinguishing between failed component inactivity and overall system survival is essential. 
The following example demonstrates the practical computation of extropy based on Equation \eqref{18}.
 \begin{example}\label{example1-livecomponent}
Consider a coherent system characterized by the signature vector
  $\pmb s =(0,\frac{1}{6},\frac{12}{7},\frac{1}{4})$, 
  which consists of four statistically IID components. Each component lifetime follows a Weibull distribution with shape parameter $\alpha>0$ and scale parameter $\lambda>0$.
Its PDF is defined as 
   $g(y)
= \frac{\alpha}{\lambda}
\left(\frac{y}{\lambda}\right)^{\alpha-1}
\exp\left[-\left(\frac{y}{\lambda}\right)^\alpha\right],
~ y>0.$
   As is well known, the reversed hazard rate function of the Weibull distribution is
    monotonically decreasing when the shape parameter satisfies $\alpha \le 1$.
For $\alpha > 1$, the reversed hazard rate function exhibits a unimodal behavior, increasing initially and then decreasing after attaining a unique maximum value.
   According to the structure induced by the signature, the system lifetime is given by
   $T(4) =\max(Y_1, \min(Y_2,Y_3), \min(Y_3,Y_4))$.
    In what follows, by applying Equation \eqref{18}, we proceed to derive the extropy associated with  the inactivity times of failed components of this system. For the present illustration, we fix the auxiliary parameters as $n=4$, $i=2$, $j=2$, and $\lambda=1.5$. Under these settings, the functional form $J(IT_{2,4}(t))$ can be derived analytically for the Weibull model.
Specifically, for the two distinct shape parameters $\alpha=0.5$ and $\alpha=2$, one obtains the following explicit closed-form representations as
% Assumptions
% i=1 j=1 n=4 s=(0 1/6 1/3 1/2)
% G(x)=1-(1+2x)e^{-2x}
\begin{equation}
J(IT_{2,4}(t))
= -\frac{181500 + 23100\big(e^{(\tfrac{t}{1.5})^{0.5}}-1\big)
+1176\big(e^{(\tfrac{t}{1.5})^{0.5}}-1\big)^2}
{10\Big(165 + 14\big(e^{(\tfrac{t}{1.5})^{0.5}}-1\big)\Big)^2},
\end{equation}
and
\begin{equation}
J(IT_{2,4}(t))
= -\frac{181500 + 23100\big(e^{(\tfrac{t}{1.5})^{2}}-1\big)
+1176\big(e^{(\tfrac{t}{1.5})^{2}}-1\big)^2}
{10\Big(165 + 14\big(e^{(\tfrac{t}{1.5})^{2}}-1\big)\Big)^2},
\end{equation}
respectively. 
 \begin{figure}[H] 
\centering
\vspace*{-0.6cm}
\includegraphics[scale=0.7]{example1-livecomponent.eps} 
\vspace*{-0.5cm}\caption{ \footnotesize{
Extropy values $J\left(IT_{2,4}(t) \right)$ of inactivity times of failed components of  system with lifetime $T(4) =\max(Y_1, \min(Y_2,Y_3), \min(Y_3,Y_4))$ in Example \ref{example1-livecomponent}.
}}\label{figexa1}
\end{figure}
   \end{example}
  The plotted curve in Figure \ref{figexa1}  represents the ٍextropy of inactivity times of failed components in the coherent system $T(4) =\max(Y_1, \min(Y_2,Y_3), \min(Y_3,Y_4))$  for the Weibull distribution with $\lambda = 1.5$ and two shape parameters $\alpha=0.5$ and $\alpha=2$.
For $\alpha=0.5$, the extropy of inactivity times of failed components increases gradually as time $t$ increases, indicating that the uncertainty associated with the inactive periods of the system components grows progressively. This behavior reflects the high variability of component lifetimes: when components follow a heavy-tailed distribution, longer inactivity periods are more unpredictable, leading to a gradual increase in extropy.
For$\alpha = 2$, the extropy initially increases as time $t$ grows, indicating a temporary rise in the uncertainty associated with the inactivity periods of the system components. This initial increase can be attributed to the overlapping or near-simultaneous inactive periods of multiple components, which momentarily intensify the overall uncertainty. After reaching a local maximum around $t \approx 2.5$, the extropy begins to decline and eventually stabilizes. This subsequent decrease and stabilization reflect the fact that, as time progresses, the inactivity periods become more predictable, leading to a reduction in uncertainty within the system.
  
 

 \begin{example}\label{example2-livecomponent}
 Consider a coherent system characterized by the signature vector
$\pmb s =(0,\ldots,0,1)$
consisting of $n$ statistically IID components. Each component lifetime follows a Weibull distribution with PDF $g(y)
= \frac{\alpha}{\lambda}
\left(\frac{y}{\lambda}\right)^{\alpha-1}
\exp\left[-\left(\frac{y}{\lambda}\right)^\alpha\right],
~ y>0.$. In this example the system is a parallel system. For a parallel system the system remains operational as long as at least one component is alive; hence, the system lifetime is the maximum of component lifetimes, i.e., $T=\max\{Y_1,\dots,Y_n\}=Y_{n:n}$. With this signature, we have $P(T=Y_{k:n})=0$ for $k=1,2,\ldots,n-1$, meaning that none of the smaller order statistics can cause system failure. Therefore, when the system is still functioning at an observation time $t>0$ (i.e., $T>t$), it is possible that several components have already failed, i.e., $Y_{k:n}<t$ for some $k<n$. In this setting, the conditional random variable of interest is the inactivity time of such failed components, defined by
$
 t - Y_{k:n}\;\big|\;\big(T>t,\,Y_{k:n}<t\big)$ for $ k=1,\ldots,n-1.
$
This variable represents the elapsed time since the failure of the $k$-th order component up to the observation epoch $t$, given that the system is still alive at time $t$. The general functional associated with the inactivity-time process is expressed, following Equation~\eqref{15} for $j=n-2$, $\lambda = 1.5$, and $\alpha=0.5$, as
\begin{eqnarray}\label{eqn}
J(IT_{n-2,n}(t)) &= &
-\frac{1}{2}(n-2)^2
\Bigg[
\frac{(n(n-1)/2)^2}
{\left( n(n-1)/2 + n(e^{(t/1.5)^{0.5}} - 1) \right)^2 (2n-5)}\nonumber\\
&+&
\frac{2 (n(n-1)/2) (n-1)n (e^{(t/1.5)^{0.5}} - 1)}
{\left( n(n-1)/2 + n(e^{(t/1.5)^{0.5}} - 1) \right)^2 (2n-5)(2n-4)}
\nonumber\\
&+&
\frac{2 ((n-1)n)^2 (e^{(t/1.5)^{0.5}} - 1)^2}
{\left( n(n-1)/2 + n(e^{(t/1.5)^{0.5}} - 1) \right)^2 (2n-5)(2n-4)(2n-3)}
\Bigg]
\end{eqnarray}

 \begin{figure}[H] 
\centering
\vspace*{-0.6cm}
\includegraphics[scale=0.7]{example2-livecomponent.eps} 
\vspace*{-0.5cm}\caption{ \footnotesize{
Behavior of  $ J(IT_{n-2,n}(t))$ as a function of $t$  for different system sizes ($n = 5, 15, 20, 25$). The plot shows that extropy decreases as the number of components increases, reflecting higher uncertainty in the inactivity times of failed components of larger systems.
}}\label{figexa2}
\end{figure}

Figure \ref{figexa2} depicts the temporal evolution of the function $J(IT_{n-2,n}(t))$ for systems of various sizes, namely $n = 5, 15, 20,$ and $25$. This measure captures the degree of uncertainty associated with system behavior, where smaller (more negative) values of $J(IT_{n-2,n}(t))$ imply greater ambiguity in identifying failed components. As the system size $n$ grows, $J(IT_{n-2,n}(t))$ consistently attains lower values, revealing that larger systems experience higher uncertainty. This tendency arises because, in extensive systems, each failure leaves many components still active, whose potential inactivity spans a wider range of durations, thereby dispersing the probability mass more broadly. Consequently, $J(IT_{n-2,n}(t))$ becomes increasingly negative, mirroring heightened unpredictability. Over time, the curves show a monotonic decline of $J(IT_{n-2,n}(t))$, with their initial slope indicating how rapidly uncertainty accumulates. Smaller systems stabilize sooner at higher (less negative) levels, whereas larger systems descend more steeply, underscoring the pronounced amplification of uncertainty with system size.

   \end{example}
   
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{eqnarray*}
%\xi J\left( IT_{j,n}(t) \right)=- \frac{1}{2}\int_0^t
%  \Big(\sum\limits_{k=i}^{n}q_k(t) \frac{\sum\limits_{l=j}^{k-1}\bar{G}_{IY^t_{l-j+1:l}}(y)\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l}{\sum\limits_{m=j}^{k-1}\binom{n}{m}(\frac{G(t)}{\bar{G}(t)})^m}\Big)^2~dy
%\end{eqnarray*} 
%   where
 %  \begin{eqnarray}\label{16}
%\bar{G}_{IY^t_{l-j+1:l}}(y) =\int_{0}^{\frac{{G}(t-y)}{{G}(t)}}j\binom{l}{j}u^{j-1}(1-u)^{l-j}~du.
%\end{eqnarray}
%and 
%\begin{eqnarray}
%q_k(t) = 
% \frac{s_k\sum\limits_{l=j}^{k-1}\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l}{\sum\limits_{m=i}^{n}s_m\sum\limits_{l=j}^{m-1}\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l},
%\end{eqnarray}
%\begin{eqnarray}
%r(t)=\frac{4ye^{-2y}}{ 1 - (1 + 2y)e^{-2y}}
%\end{eqnarray}
%$g(y) = 4ye^{-2y},~y > 0$, and the CDF $G(y) = 1 - (1 + 2y)e^{-2y}$
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   
%Using Equation \eqref{15}, one can systematically evaluate the extropy index for the inactivity times of two failed components, facilitating a structured comparison across diverse systems. As an initial step, it is instructive to revisit the concepts of various stochastic orderings, which form the theoretical backbone for such comparative analyses. These ideas have been thoroughly explored in the literature, notably in the works of Shaked and Shanthikumar (2007) and Belzunce et al. (2015).
   %\begin{equation}
 %  f(t)=\frac{ \binom{n}{l}t^l}{\sum\limits_{l=k}^{j-1}\binom{n}{m}t^m}
 %  \end{equation}
% \begin{definition} {\bf (Stochastic orders)} Let $X$ and $Y$ be two random variables with CDFs $F$ and $G$ and PDFs $f$ and $g$, respectively.   Then
% $X$ is said to be smaller than $Y$  in the sense of: \\
%{{(i)}} usual stochastic order (denoted  by $X\leqslant_{st}Y$  {or $F\leqslant_{st}G$}) if $ \bar{F}(x)\leq \bar{G}(x)$ for all $x$;\\
%{{(ii)}} 
 %reversed hazard rate order (denoted by $X\leqslant_{rhr}Y$  {or $F\leqslant_{hr}G$}) if  ${\displaystyle  \frac{{G}(x)}{{F}(x)}}$ is increasing in $x$.\\
%{{(iii)}} 
%disperse order (denoted by $X\leqslant_{disp}Y$  {or $F\leqslant_{disp}G$}) if 
%$g(G^{-1}(v))\leq f(F^{-1}(v))$ for all $0<v<1$.\\
%Also, let $\boldsymbol{ p}$ and $\boldsymbol{ q}$
 %  be  two discrete distributions on the integers $\{1, ..., n\}$. Then, it is said that   (see, for example, Kochar et al. 1999) \\
%{{(iv)}}  $\boldsymbol{ p}\leqslant _{st}\boldsymbol{ q}$ if and only if $\sum\limits_{i=j}^{n}p_{i}\leq \sum\limits_{i=j}^{n}q_{i}$, for $j=1,\cdots,n$.\\
%{{(v)}}  ${\bf p}\leqslant _{rhr}{\bf q}$ if and only if ${\sum\limits_{i=1}^{j}p_{i}}\Big/{\sum\limits_{i=1}^{j}q_{i}}$ is decreasing in $j$, for  $j=1,\cdots,n$.\\
%{{(vi)}}   ${\bf p}\leqslant _{lr}{\bf q}$ if and only if ${p_{i}}/q_{i}$ is decreasing in $i$, for  $i=1,\cdots,n$ when $p_{i},q_{i}>0$.
%{{(v)}}  ${\bf p}\leqslant _{hr}{\bf q}$ if and only if ${\sum\limits_{i=j}^{n}p_{i}}\Big/{\sum\limits_{i=j}^{n}q_{i}}$ is decreasing in $j$, for  $j=1,\cdots,n$.\\
%{{(vi)}}   ${\bf p}\leqslant _{lr}{\bf q}$ if and only if ${p_{i}}/q_{i}$ is decreasing in $i$, for  $i=1,\cdots,n$ when $p_{i},q_{i}>0$.
%\end{definition}
%In the subsequent development, we formalize the system’s model function as
%$${\cal{M}}_{IT_{j,n}(t), IY_{j,k,n}(t), IY^t_{l-j+1:l},\boldsymbol{q}(t),Y,G}=\{IT_{j,n}(t), IY_{j,k,n}(t), IY^t_{l-j+1:l},\boldsymbol{q}(t),Y,G \},
%$$
%where the system under consideration is composed of $n$ IID components with lifetimes
 %$Y_1, \ldots, Y_n$
%each governed by a common CDF $G$.
%The random variable $IT_{j,n}(t)$ denotes the inactivity time associated with components that have failed by time $t$. The vector
%$\pmb q(t) = (0, \ldots , 0,q_i(t),\ldots, q_n(t))$
%collects the weighting coefficients as specified in Equation \eqref{13}. 
%Moreover, we assume that the random variable $Y$  corresponds to the lifetime of one of the individual components, i.e., $Y_1, \ldots, Y_n$.

%In reliability analysis, extropy offers a complementary lens to entropy for quantifying uncertainty in system behavior. It focuses on the information carried by low-probability events, capturing aspects of system unreliability that conventional dispersion or entropy-based measures may overlook. Lower extropy values imply higher disorder and unpredictability in the inactivity of components, whereas higher values indicate greater regularity and informational coherence in system response after failures. By emphasizing the structure of rare events, extropy provides engineers and analysts with a refined tool to identify hidden patterns of dependence and subtle irregularities in failure dynamics. This insight not only enhances the probabilistic understanding of system performance but also supports more informed decisions in maintenance planning, risk management, and reliability optimization. Ultimately, extropy serves as both a quantitative index of uncertainty and a qualitative descriptor of the informational symmetry underlying system robustness.



%The following result formalizes how the usual stochastic, hazard rate, and likelihood-ratio orderings between the coefficient vectors of two system’s model functions translate into an ordering of the extropy index for the corresponding inactivity times.








%\begin{theorem}\label{theorem1}
%Consider ${\cal{M}}_{IT^Y_{j,n}(t), IY_{j,k,n}(t), IY^t_{l-j+1:l},\boldsymbol{q^Y}(t),Y,G}$ and %${\cal{M}}_{IT^X_{j,n}(t), IX_{j,k,n}(t), IX^t_{l-j+1:l},\boldsymbol{q^X}(t),X,G}$ as the
%model functions for inactivity times of failed components of two distinct systems. Moreover, assume that the difference ${G}_{IT^Y_{j,n}(t)}(y)-{G}_{IT^X_{j,n}(t)}(y)$ is an increasing function with respect to $y$.
%\begin{itemize}
%\item[(i)] If 
% ${\boldsymbol{ q}}^Y(t)\leqslant_{st}{\boldsymbol{q}}^{X}(t)$, then $  J(IT^Y_{j,n}(t))\leq  J(IT^X_{j,n}(t))$.
% \item[(ii)] If
% ${\boldsymbol{ q}}^{Y}(t)\leqslant_{rhr}{\boldsymbol{q}}^{X}(t)$, then $  J(IT^Y_{j,n}(t))\leq  J(IT^X_{j,n}(t))$.
%  \item[(iii)] 
% If 
%${\boldsymbol{ q}}^{Y}(t)\leqslant_{lr}{\boldsymbol{q}}^{X}(t)$,  then $  J(IT^Y_{j,n}(t))\leq  J(IT^X_{j,n}(t))$.
%\end{itemize}
%\end{theorem}
%{\bf{Proof.}} 
%The derivation of these results follows in a straightforward manner from the foundational arguments established in Theorem 3.3 of Goliforushani et al. (2012), by employing the assumption that the difference
% ${G}_{IT^Y_{j,n}(t)}(y)-{G}_{IT^X_{j,n}(t)}(y)$
%is an increasing function with respect to $y$.
%$\hfill \Box$

%The subsequent theorem formalizes a key monotonicity result concerning the extropy measure when the underlying lifetime distribution possesses an decreasing reversed hazard rate.






%\begin{theorem}
%Define the reversed hazard rate as
%$
%r(t)=\frac{f(t)}{G(t)}}.
%$
%Let
%$
%a(t)=\frac{G(t-y)}{G(t)}$ and
%$\phi(u)=j\binom{l}{j}u^{j-1}(1-u)^{l-j}$.
%Therefore, from \eqref{12}, we have 
%$
%\overline{G}_{IY^t_{l-j+1:l}}(y)=\int_{0}^{a(t)}\phi(u)du.
%$
%Then for every $t$ where the derivatives exist, we have
%\begin{equation}\label{eq:second-derivative}
%\frac{d^2}{dt^2}\overline{G}_{IY^t_{l-j+1:l}}(y)
%=\phi(a(t))a(t)\left[
%\big(r(t-y)-r(t)\big)^2\frac{1-(l-1)a(t)}{1-a(t)} + \big(r'(t-y)-r'(t)\big)
%\right].
%\end{equation}
%Consequently, the second derivative is positive (resp. negative) if and only if the following term is positive (resp.  negative); i.e.
%\[
%\frac{d^2}{dt^2}\overline{G}_{IY^t_{\,l-j+1:l}}(y)>0
%\quad\Longleftrightarrow\quad
%\big(r(t-y)-r(t)\big)^2\frac{1-(l-1)a(t)}{1-a(t)} + \big(r'(t-y)-r'(t)\big) >0,
%\]
%and similarly with ``\(<0\)'' for negativity.
%\end{theorem}

%{\bf{Proof.}} 
%We write \(a=a(t)\) and \(\Delta r:=r(t-y)-r(t)\) for brevity.
% By the fundamental theorem of calculus and the chain rule,
%\[
%\frac{d}{dt}\overline{G}_{IY^t_{\,l-j+1:l}}(y)=\phi(a)a'(t).
%\]
%Differentiating \(\ln a(t)=\ln G(t-y)-\ln G(t)\) gives
%$
%\frac{a'(t)}{a(t)}=\frac{G'(t-y)}{G(t-y)}-\frac{G'(t)}{G(t)}=r(t-y)-r(t)=\Delta r,
%$
%hence
%\[
%a'(t)=a\Delta r,
%\qquad
%\frac{d}{dt}\overline{G}(y)=\phi(a)\,a\,\Delta r.
%\]

 %Differentiate the first derivative:
%\[
%\frac{d^2}{dt^2}\overline{G}(y)
%=\frac{d}{dt}\big(\phi(a)\,a'(t)\big)
%=\phi'(a)\,a'(t)^2 + \phi(a)\,a''(t).
%\]
%We already have \(a'(t)=a\Delta r\). Differentiate \(a'(t)=a\Delta r\) to obtain
%\[
%a''(t)=\frac{d}{dt}\big(a\Delta r\big)
%= a'(t)\Delta r + a(t)\frac{d}{dt}\big(r(t-y)-r(t)\big).
%\]
%Since \(\frac{d}{dt}(r(t-y)-r(t))=r'(t-y)-r'(t)\) and \(a'(t)=a\Delta r\), we get
%\[
%a''(t)=a\Delta r^2 + a\big(r'(t-y)-r'(t)\big)
%= a\Big(\Delta r^2 + r'(t-y)-r'(t)\Big).
%\]

%Substitute \(a'\) and \(a''\) into the expression for the second derivative:
%\[
%\begin{aligned}
%\frac{d^2}{dt^2}\overline{G}(y)
%&= \phi'(a)\,a^2\Delta r^2 + \phi(a)\,a\Big(\Delta r^2 + r'(t-y)-r'(t)\Big)\\[4pt]
%&= \phi(a)\,a\left[\Delta r^2\Big(1 + a\frac{\phi'(a)}{\phi(a)}\Big) + \big(r'(t-y)-r'(t)\big)\right].
%\end{aligned}
%\]
%Now compute \(\phi'(u)\). Since \(\phi(u)=j\binom{l}{j}u^{j-1}(1-u)^{\,l-j}\),
%\[
%\frac{\phi'(u)}{\phi(u)}=\frac{j-1}{u}-\frac{l-j}{1-u}.
%\]
%Thus
%\[
%1 + a\frac{\phi'(a)}{\phi(a)}
%=1 + a\Big(\frac{j-1}{a}-\frac{l-j}{1-a}\Big)
%=\frac{1-(l-1)a}{1-a}.
%\]
%Substituting this into the previous display yields exactly equation \eqref{eq:second-derivative}:
%\[
%\frac{d^2}{dt^2}\overline{G}(y)
%=\phi(a)\,a\left[
%\Delta r^2\frac{1-(l-1)a}{1-a} + \big(r'(t-y)-r'(t)\big)
%\right].
%\]

%Finally, since \(\phi(a)>0\) and \(a>0\) for \(a\in(0,1)\), the sign of the left-hand side equals the sign of the bracketed expression. This gives the stated necessary-and-sufficient conditions for positivity or negativity of the second derivative.
%$\hfill \Box$








%\begin{theorem}\label{theorem3}
%Consider the model function ${\cal{M}}_{IT^Y_{j,n}(t), IY_{j,k,n}(t), IY^t_{l-j+1:l},\boldsymbol{q}(t),Y,G}$, which represents the inactivity times of failed components within a coherent system. Let $Y$ be a non-negative random variable with SF $\bar{G}(t)$ and PDF $g(t)$. Then the CRJ measure $\xi J(I T_{j,n}(t))$ is also decreasing with respect to $t \geq 0$.
%\end{theorem}
%{\bf{Proof.}} 
%Let
%\begin{eqnarray}
%\pi_{j,k,n}(t,u)=\sum\limits_{k=i}^{n}\sum\limits_{l=j}^{k-1} \frac{s_k \binom{n}{l} (\frac{G(t)}{\bar{G}(t)})^l}{\sum\limits_{m=i}^{n}\sum\limits_{l=j}^{m-1} s_m \binom{n}{l} (\frac{G(t)}{\bar{G}(t)})^l}j\binom{l}{j}(1-u)^{j-1}u^{l-j}.
%\end{eqnarray}
%Therefore, we have
%\begin{eqnarray}
%\frac{d}{dt}\pi_{j,k,n}(t,u)=\sum\limits_{k=i}^{n}\sum\limits_{l=j}^{k-1} \Bigg(B_{l,j,k,n}(t)\frac{d}{dt}q_k(t) +q_k(t)\frac{d}{dt} B_{l,j,k,n}(t)\Bigg)j\binom{l}{j}(1-u)^{j-1}u^{l-j}.
%\end{eqnarray}
%Suppose that $V_l(t)=\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l$ and $R_{j,k}(t)=\sum\limits_{l=j}^{k-1}\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l$. Let $\phi(t)=\frac{G(t)}{\bar{G}(t)}$. Since $\phi(t)$ is an increasing function of $t$, without loss of generality, we can consider $V_l(t)=\binom{n}{l}t^l$ and $R_{j,k}(t)=\sum\limits_{l=j}^{k-1}\binom{n}{l}t^l$.  Therefore, we obtain
%\begin{eqnarray}
%\frac{d}{dt}q_k(t)=\frac{\sum\limits_{m=i}^{n}s_ks_m\Big(R^\prime_{j,k}(t)R_{j,m}(t)-R_{j,k}(t)R^\prime_{j,m}(t) \Big)}{\Big( \sum\limits_{m=i}^{n}s_mR_{j,m}(t)\Big)^2},
%\end{eqnarray}
%and
%\begin{eqnarray}
%\frac{d}{dt}B_{l,j,k,n}(t)=\frac{\sum\limits_{m=j}^{k-1}\Big(V^\prime_l(t)V_m(t)-V_l(t)V^\prime_m(t)\Big)}{\Big( \sum\limits_{m=j}^{k-1}V_m(t)\Big)^2}.
%\end{eqnarray}
%By doing some calculation, we have
%\begin{eqnarray}
% R_{j,k}'(t) R_{j,m}(t) - R_{j,k}(t) R_{j,m}'(t)
%&=& \left\{ \sum_{l=j}^{k-1} \binom{n}{l} t^{l}\right\}'
%     \left\{ \sum_{l=j}^{m-1} \binom{n}{l} t^{l} \right\}
% - \left\{ \sum_{l=j}^{k-1} \binom{n}{l} t^{l} \right\}
  %   \left\{ \sum_{l=j}^{m-1} \binom{n}{l} t^{l} \right\}' \nonumber\\
%&=&
  %   \sum_{l_1 = j}^{k-1} \sum_{l_2 = j}^{m-1}
  %   l_1 \binom{n}{l_1} \binom{n}{l_2}
  %   t^{l_1 + l_2 - 1}
 %    - 
 %    \sum_{l_1 = j}^{k-1} \sum_{l_2 = j}^{m-1}
  %   l_2 \binom{n}{l_1} \binom{n}{l_2}
  %   t^{l_1 + l_2 - 1}  \nonumber\\
% &=& 
 %    \sum_{l_1 = j}^{k-1} \sum_{l_2 = j}^{m-1}
 %    \binom{n}{l_1} \binom{n}{l_2}
%     t^{l_1 + l_2 - 1} (l_1 - l_2) 
% \geq  0.
 %\end{eqnarray}
%and
 %\begin{eqnarray}
% V^\prime_l(t)V_m(t)-V_l(t)V^\prime_m(t)= \binom{n}{l}\binom{n}{m}(l-m)t^{l+m-1}\leq 0
% \end{eqnarray}
% $\hfill \Box$


%\begin{equation}
%\begin{aligned}
%\frac{\partial}{\partial t} {g}_{IY^t_{l-j+1:l}}(y) 
%= j \binom{l}{j} & \Bigg[
%\frac{g'(t-y) G(t) - g(t-y) g(t)}{G(t)^2} 
%\left(\frac{G(t-y)}{G(t)}\right)^{j-1} 
%\left(1 - \frac{G(t-y)}{G(t)}\right)^{l-j} \\
%& + \frac{g(t-y)}{G(t)} 
%\left(\frac{G(t-y)}{G(t)}\right)^{j-2} 
%\left(1 - \frac{G(t-y)}{G(t)}\right)^{l-j-1} 
%\left[(j-1) - (l-1)\frac{G(t-y)}{G(t)}\right] 
%\frac{g(t-y) G(t) - G(t-y) g(t)}{G(t)^2} 
%\Bigg].
%\end{aligned}
%\end{equation}










In what follows, our objective is to establish a rigorous lower bound for the extropy measure associated with the inactivity times of failed components. This bound is formalized in the subsequent theorem, which expresses it in terms of the extropy corresponding to the inactivity times in an $(n-k+1)$-out-of-$n$ coherent system. Specifically, the formulation applies under the condition that the system remains operational at time $t$, while the component characterized by the lifetime $X_{k:n}$ has experienced failure at some instant prior to $t$, with $k$ ranging from 1 to $i$. 

\begin{theorem}\label{theorem4}
 Let $J(IT_{j,n}(t))$ denote the Extropy of the inactivity times of failed components, corresponding to the model function
 ${\cal{M}}_{IT_{j,n}(t), IY_{j,k,n}(t), IY^t_{l-j+1:l},\boldsymbol{q}(t),Y,G}$.  Then, the following inequality holds 
 \begin{equation}
  J(IT_{j,n}(t))\geq\sum\limits_{r=i}^{n}q_r(t)  J(IY_{j,r,n}(t)),
 \end{equation}
where, $IY_{j,r,n}(t)$ is  the inactivity time of failed components in an $(n-r+1)$-out-of-$n$ system with uniformly distributed components, when
the system is working at time $t$ and the component with lifetime $X_{k:n}$ has failed
sometime before  $t$ for $k = 1, \ldots, i$.
\end{theorem}
{\bf{Proof.}}
Invoking Jensen’s inequality on equation \eqref{15} allows us to derive
\begin{eqnarray*}
 J(IT_{j,n}(t))&\geq &
- \frac{1}{2}\sum\limits_{r=i}^{n} q_r(t)\int_0^\infty \Big( g_{IY_{j,r,n}(t)}(y)\Big)^2~dy\\&=&
\sum\limits_{r=i}^{n}q_r(t)   J(IY_{j,r,n}(t)).
\end{eqnarray*}
Equality is realized in  a $(n-k+1)$-out-of-$n$ systems scenario, as the weighting function assigns $q_{r}(t)=0$ for all $r$ not equal to $(n-k+1)$ and $q_{r}(t)=1$ when $r= n-k+1$.
$\hfill \Box$


%\begin{equation}\label{13}
%q_k(t) = \frac{ s_k \sum\limits_{l=j}^{k-1}\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l}
%{ \sum\limits_{m=i}^{n}\sum\limits_{l=j}^{m-1} s_m \binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l },
%\end{equation}
%and
%\begin{equation}\label{14}
%B_{l,j,k,n}(t) = \frac{\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l}
%{  \sum\limits_{l=j}^{k-1}\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l}.
%\end{equation}

%where
%\begin{equation}\label{11pdf}
%{g}_{IY_{j,k,n}(t)}(y) = \sum\limits_{l=j}^{k-1} B_{l,j,k,n}(t) {g}_{IY^t_{l-j+1:l}}(y),
%\end{equation}
%and 
%\begin{equation}\label{12pdf}
%{g}_{IY^t_{l-j+1:l}}(y) =j\binom{l}{j}\frac{{g}(t-y)}{{G}(t)}(\frac{{G}(t-y)}{{G}(t)})^{j-1}(1-\frac{{G}(t-y)}{{G}(t)})^{l-j},~0<y<t.
%\end{equation}

%\begin{eqnarray}\label{15}
%J\left( IT_{j,n}(t) \right) =- \frac{1}{2}\int_0^1 \Big(\sum\limits_{k=i}^{n}\sum\limits_{l=j}^{k-1} q_k(t) B_{l,j,k,n}(t)j\binom{l}{j}(1-u)^{j-1}u^{l-j}\Big)^2~du.
%\end{eqnarray}

%\begin{eqnarray*}
% J(IT_{j,n}(t))&\geq &
%- \frac{1}{2}\sum\limits_{r=i}^{n} q_r(t)\int_0^\infty \Big( g_{IY_{j,r,n}(t)}(y)\Big)^2~dy
%\end{eqnarray*}


%\begin{eqnarray*}%
% J(IT_{j,n}(t))&\geq &
% - \frac{1}{2}\sum\limits_{k=i}^{n}\sum\limits_{l=j}^{k-1} q_k(t) B_{l,j,k,n}(t)\int_0^\infty\Big(g_{IY^t_{l-j+1:l}}(y)\Big)^2~dy
%\end{eqnarray*}
%First bound is
%\begin{eqnarray*}
%\xi J(IT(n,t))\geq
%\sum\limits_{r=1}^{i}p_r(t) \Bigg[- \frac{1}{2}\int_0^\infty \Big( \frac{\sum\limits_{l=k}^{j-1}\bar{G}_{Y^t_{j-1,n-l}}(y)\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l}{\sum\limits_{m=k}^{j-1}\binom{n}{m}(\frac{G(t)}{\bar{G}(t)})^m}\Big)^2~dy\Bigg].
%\end{eqnarray*}
%where
%\begin{eqnarray}\label{16}
%\bar{G}_{Y^t_{j-1,n-l}}(y) =1-(j-l)\binom{n-l}{n-j}\int_{\frac{\bar{G}(y+t)}{\bar{G}(t)}}^{1}u^{n-j}(1-u)^{j-l-1}~du.
%\end{eqnarray}
%\begin{eqnarray}
%p_k(t) = 
% \frac{s_k\sum\limits_{l=k}^{j-1}\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l}{\sum\limits_{m=1}^{i}s_m\sum\limits_{l=m}^{j-1}\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l},
%\end{eqnarray}
%and secound bound is 
%\begin{eqnarray*}
%\xi J(IT(n,t))\geq 
 %\sum\limits_{k=1}^{i}\sum\limits_{l=k}^{j-1} p_k(t) K_{l,j,k,n}(t)\Bigg[- \frac{1}{2}\int_0^\infty \Big( 
 %1-(j-l)\binom{n-l}{n-j}\int_{\frac{\bar{G}(y+t)}{\bar{G}(t)}}^{1}u^{n-j}(1-u)^{j-l-1}~du
 %\Big)^2~dy\Bigg]
%\end{eqnarray*}
%where
%\begin{eqnarray}
%K_{l,j,k,n}(t) = 
% \frac{\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l}{\sum\limits_{m=k}^{j-1}\binom{n}{m}(\frac{G(t)}{\bar{G}(t)})^m},
%\end{eqnarray}

The following theorem provides an upper bound for the extropy  of the inactivity times of failed components in the coherent system, expressed in terms of the weighted sum of the extropy  of the shifted inactivity variables $IY^t_{l-j+1:l}$, with weights determined by $q_k(t) B_{l,j,k,n}(t)$.

\begin{theorem}\label{theorem5}
 Let $  J(IT_{j,n}(t))$ denote the extropy of the inactivity times of failed components with the corresponding model function
 ${\cal{M}}_{IT_{j,n}(t), IY_{j,k,n}(t), IY^t_{l-j+1:l},\boldsymbol{q}(t),Y,G}$.  Then 
 \begin{equation}
  J(IT_{j,n}(t))\geq\sum\limits_{k=i}^{n}\sum\limits_{l=j}^{k-1} q_k(t) B_{l,j,k,n}(t)  J(IY^t_{l-j+1:l}).
 \end{equation}
\end{theorem}
{\bf{Proof.}}
Through the application of Jensen’s inequality to equation \eqref{15}, the following result is obtained
\begin{eqnarray*}
 J(IT_{j,n}(t))&\geq &
 - \frac{1}{2}\sum\limits_{k=i}^{n}\sum\limits_{l=j}^{k-1} q_k(t) B_{l,j,k,n}(t)\int_0^\infty\Big(g_{IY^t_{l-j+1:l}}(y)\Big)^2~dy\\&=&
\sum\limits_{k=i}^{n}\sum\limits_{l=j}^{k-1} q_k(t) B_{l,j,k,n}(t)  J(IY^t_{l-j+1:l}).
\end{eqnarray*}
$\hfill \Box$

%\begin{eqnarray}
%\xi J(IT(n,t))= 
% - \frac{1}{2}\int_0^\infty\Big(\sum\limits_{k=1}^{i} p_k(t) 
 %\frac{\sum\limits_{l=k}^{j-1}\bar{G}_{Y^t_{j-1,n-l}}(y)\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l}{\sum\limits_{m=k}^{j-1}\binom{n}{m}(\frac{G(t)}{\bar{G}(t)})^m}\Big)^2~dy,
%\end{eqnarray}
%where
%\begin{eqnarray}\label{16}
%\bar{G}_{Y^t_{j-1,n-l}}(y) =1-(j-l)\binom{n-l}{n-j}\int_{\frac{\bar{G}(y+t)}{\bar{G}(t)}}^{1}u^{n-j}(1-u)^{j-l-1}~du.
%\end{eqnarray}
%and
%\begin{eqnarray}
%p_k(t) = 
% \frac{s_k\sum\limits_{l=k}^{j-1}\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l}{\sum\limits_{m=1}^{i}s_m\sum\limits_{l=m}^{j-1}\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l},
%\end{eqnarray}
In the example below, we analyze and compare the lower bounds provided by Theorems \ref{theorem4} and \ref{theorem5}.


%\begin{eqnarray*}
%JCRJ(IT_{j,n}(t):IY_{j,i,n}(t),\ldots, IY_{j,n,n}(t))=- \frac{1}{2}\int_0^t
%  \Big(\sum\limits_{k=i}^{n}q_k(t) \frac{\sum\limits_{l=j}^{k-1}\bar{G}_{IY^t_{l-j+1:l}}(y)\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l}{\sum\limits_{m=j}^{k-1}\binom{n}{m}(\frac{G(t)}{\bar{G}(t)})^m}\Big)^2~dy-
%   \sum\limits_{r=i}^{n} q_r(t)\Big[
%- \frac{1}{2}\int_0^\infty \Big( \bar{G}_{IY_{j,r,n}(t)}(y)\Big)^2~dy\Big]\\
%\end{eqnarray*} 


%\begin{eqnarray*}
%JCRJ(IT_{j,n}(t):IY_{j,i,n}(t),\ldots, IY_{j,n,n}(t))=\frac{1}{2}\sum\limits_{k=i}^{n}q_k(t)\int_{0}^{t}
%\Big(\bar{G}_{IT_{j,n}(t)}(y)-\bar{G}_{IY_{j,k,n}(t)}(y)\Big)^2~dy
%\end{eqnarray*}

%\begin{equation}\label{10}
%\bar{G}_{IT_{j,n}(t)}(y) = \sum\limits_{k=i+1}^{n} q_k(t) \bar{G}_{IY_{j,k,n}(t)}(y),
%\end{equation}


 %  where
  % \begin{eqnarray}\label{16}
%\bar{G}_{IY^t_{l-j+1:l}}(y) =1-\int_{\frac{{G}(t-y)}{{G}(t)}}^{1}j\binom{l}{j}u^{j-1}(1-u)^{l-j}~du.
%\end{eqnarray}
%and 
%\begin{eqnarray}
%q_k(t) = 
% \frac{s_k\sum\limits_{l=j}^{k-1}\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l}{\sum\limits_{m=i}^{n}s_m\sum\limits_{l=j}^{m-1}\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l},
%\end{eqnarray}

%\begin{equation}\label{11}
%\bar{G}_{IY_{j,k,n}(t)}(y) = \sum\limits_{l=j}^{k-1} B_{l,j,k,n}(t)\,\bar{G}_{IY^t_{l-j+1:l}}(y),
%\end{equation}

%\begin{equation}\label{14}
%B_{l,j,k,n}(t) = \frac{ \binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l}
%{ \sum\limits_{l=j}^{k-1}\binom{n}{l}(\frac{G(t)}{\bar{G}(t)})^l}.
%\end{equation}

%\begin{eqnarray*}
%\xi J(IT_{j,n}(t))= \sum\limits_{r=i}^{n} q_r(t)\Big[
%- \frac{1}{2}\int_0^\infty \Big( \bar{G}_{IY_{j,r,n}(t)}(y)\Big)^2~dy\Big]\\
%\end{eqnarray*}


%\begin{eqnarray*}
%\xi  J(IT_{j,n}(t))\geq \sum\limits_{k=i}^{n}\sum\limits_{l=j}^{k-1} q_k(t) B_{l,j,k,n}(t)\Big[
% - \frac{1}{2}\int_0^\infty\Big(\bar{G}_{IY^t_{l-j+1:l}}(y)\Big)^2~dy\Big]
%\end{eqnarray*}




\begin{example}\label{example4-livecomponent}
Consider a coherent system characterized by the signature vector 
$
\pmb{s} = \left(0, \frac{1}{3}, \frac{5}{12}, \frac{1}{4}\right),
$ 
which consists of four statistically IID components. Each component lifetime follows a Weibull distribution with shape parameter \(\alpha = 0.8\) and scale parameter \(\lambda = 3\), having PDF 
$
g(y) = \frac{\alpha}{\lambda}\left(\frac{y}{\lambda}\right)^{\alpha-1} e^{-(y/\lambda)^\alpha}, \quad y>0,
$
and CDF 
$
G(y) = 1 - e^{-(y/\lambda)^\alpha}, \quad y>0.
$
After carrying out the calculations, the lower bounds stated in Theorems \ref{theorem4} and \ref{theorem5} are obtained in Figure \ref{fig11}.
In Figure \ref{fig11}, we illustrate the extropy function $J(IT_{2,4}(t))$ together with the lower bounds derived in Theorems \ref{theorem4} and \ref{theorem5}. 
The first lower bound, derived from Theorem \ref{theorem4}, is denoted by $L_1(t)$, whereas the second lower bound, established in Theorem \ref{theorem5}, is represented by $L_2(t)$.   
As shown in the Figure \ref{fig11}, the extropy function $ J(IT_{2,4}(t))$ decreases smoothly with time $t$. Both bounds provide conservative approximations, but $L_1(t)$ lies consistently closer to the true extropy curve, indicating that the bound derived in Theorem \ref{theorem4} is sharper than the one obtained in Theorem \ref{theorem5}.

\end{example}
\begin{figure}[H] 
\centering
\vspace*{-0.6cm}
\includegraphics[scale=0.7]{example4-livecomponent.eps} 
\vspace*{-0.5cm}\caption{ \footnotesize{
Comparison of the two lower bounds, $L_1(t)$ and $L_2(t)$, for the extropy of the inactivity times failed components in systems with Weibull-distributed component lifetimes. The results indicate that $L_1(t)$ consistently lies closer to the exact extropy values, thereby providing a sharper and more accurate approximation than $L_2(t)$.
  }}\label{fig11}
\end{figure}




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\section{Coherent systems of four Weibull distributed components with the signature vector $\pmb{s} = (0, \ldots, 0, s_i, \ldots, s_n)$
}
This section delves into the extropy for the inactivity times of the failed components in a coherent system composed of $n$ statistically IID units, where each component lifetime follows a Weibull distribution. The PDF and CDF of each component lifetime are given by
$g_Y(y) = \frac{k}{\lambda} \left(\frac{y}{\lambda}\right)^{k-1} e^{-(y/\lambda)^k}, \quad y>0,$
and $G_Y(y) = 1 - e^{-(y/\lambda)^k}.$
In what follows, we consider coherent systems with signature vectors of the form
$\pmb{s} = (0, \ldots, 0, s_i, \ldots, s_n)$ for $1<i\leq n$, so that only the order statistics $X_{k:n}$ with $k>i$ may represent the system’s failure time. Equivalently, 
$
P(T=X_{k:n})=0, \quad k=1,2,\ldots,i,
$
which means that the early lifetimes $X_{1:n},\ldots,X_{i:n}$ cannot trigger system failure. As a result, when the system is still operational at time $t$ (i.e., $T>t$), it is possible that these components have already failed before $t$, that is, $X_{k:n}<t$ for some $k \leq i$.  For illustration, Table~\ref{tab:coherent-systems} presents several examples of four-component coherent systems characterized by their structure function $T(4)$ and the corresponding signature vectors of the form $\pmb{s} = (0, \ldots, 0, s_i, \ldots, s_n)$. 

\begin{table}[h!]
\centering
\caption{Representative examples of coherent systems with four components, described through the structure function $T(4)$ and their associated signature vectors in the form $\pmb{s} = (0, \ldots, 0, s_i, \ldots, s_n)$. }
\label{tab:coherent-systems}
\begin{tabular}{cll}
\toprule
\textbf{System} & \textbf{$T=\phi(Y_1,Y_2,Y_3,Y_4)$} & \textbf{Signature} \\
\midrule
1 & $Y_{2:2} = \max(Y_1,Y_2)$ (2-parallel) 
  & $\left(0, \tfrac{1}{6}, \tfrac{1}{3}, \tfrac{1}{2}\right)$ \\

2 & $\max(Y_2, \min(Y_1,Y_3))$ (consecutive 2-out-of-3: $F$) 
  & $\left(0,\tfrac{1}{3}, \tfrac{5}{12}, \tfrac{1}{4}\right)$ \\

3 & $\max(Y_1, \min(Y_2,Y_3,Y_4))$ 
  & $\left(0,\tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{4}\right)$ \\

4 & $\max(Y_1, \min(Y_2,Y_3), \min(Y_3,Y_4))$ (3-parallel) 
  & $\left(0,\tfrac{1}{6}, \tfrac{7}{12}, \tfrac{1}{4}\right)$ \\

5 & $\max(Y_1,Y_2,Y_3)$ 
  & $\left(0,0,\tfrac{1}{4}, \tfrac{3}{4}\right)$ \\
  
  6 & $\max(Y_{2:3},Y_4)$ 
  & $\left(0, 0,\tfrac{3}{4}, \tfrac{1}{4}\right)$ \\

7 & $\min(\max(Y_1,Y_2,Y_3), \max(Y_2,Y_3,Y_4))$ (consecutive 3-out-of-4: $F$) 
  & $\left(0, 0,\tfrac{1}{2}, \tfrac{1}{2}\right)$ \\

8 & $Y_{4:4} = \max(Y_1,Y_2,Y_3,Y_4)$ (parallel) 
  & $\left(0, 0, 0,1\right)$ \\
\bottomrule
\end{tabular}
\end{table}


%\begin{eqnarray}
%\xi J(IT(n,t)) &=& 
% - \frac{1}{2}\int_0^\infty\Big(\sum_{k=1}^{i} p_k(t)
% \frac{\sum_{l=k}^{j-1}\bar{G}_{Y^t_{\,j-1,n-l}}(y)\binom{n}{l}\big(\frac{G(t)}{\bar{G}(t)}\big)^l}
% {\sum_{m=k}^{j-1}\binom{n}{m}\big(\frac{G(t)}{\bar{G}(t)}\big)^m}\Big)^{2}\,dy,
%\end{eqnarray}
%where
%\begin{eqnarray}\label{16}
%\bar{G}_{Y^t_{\,j-1,n-l}}(y) 
%&=& 1 - (j-l)\binom{n-l}{\,n-j\,}\int_{\frac{\bar{G}(y+t)}{\bar{G}(t)}}^{1} u^{\,n-j}(1-u)^{\,j-l-1}\,du.
%\end{eqnarray}
%and
%\begin{eqnarray}
%p_k(t) =
%\frac{s_k\displaystyle\sum_{l=k}^{j-1}\binom{n}{l}\big(\frac{G(t)}{\bar{G}(t)}\big)^l}
%{\displaystyle\sum_{m=1}^{i} s_m \sum_{l=m}^{j-1}\binom{n}{l}\big(\frac{G(t)}{\bar{G}(t)}\big)^l }.
%\end{eqnarray} 

%Due to the lack of memory property of the exponential distribution, this result is obvious.
%Let the coefficient vector in Equation \eqref{2000} be denoted by  $\boldsymbol{ q}^*(t)=(q^*_1(t), q^*_2(t),\ldots,q^*_n(t))$.
%for $0.008<t<0.048$, 
%{\scriptsize
%\begin{equation}\label{eq2}
%\xi J(T_4(3,t))\geq J(T_2(3,t))\geq \xi J(T_5(3,t))\geq \xi J(T_6(3,t))\geq \xi J(T_1(3,t))\geq \xi J(T_7(3,t))\geq \xi J(T_3(3,t))\geq \xi J(T_8(3,t)),
%\end{equation}
%}
%and for $t>0.048$,  we have
%{\scriptsize
%\begin{equation}\label{eq3}
%\xi J(T_4(3,t))\geq \xi J(T_2(3,t))\geq \xi J(T_5(3,t))\geq \xi J(T_6(3,t))\geq \xi J(T_7(3,t))\geq \xi J(T_1(3,t))\geq \xi J(T_3(3,t))\geq \xi J(T_8(3,t)).
%\end{equation}
\begin{figure}[H] 
\centering
\includegraphics[scale=0.7]{example5-livecomponent.eps} 
\vspace*{-0.5cm}\caption{ \footnotesize{
Variation of $J(IT_{2,4}(t))$ for the eight coherent systems listed in Table \ref{tab:coherent-systems}, obtained under a Weibull lifetime model with shape parameter $k=0.4$ and scale parameter $\lambda=1.5$. The results indicate that System 6 attains the smallest $ J(IT_{1,4}(t))$ values, representing the highest degree of structural uncertainty, while System 3 achieves the largest $\xi J(IT_{1,4}(t))$ values, corresponding to the most reliable and least uncertain configuration among the examined systems. 
}}
\label{fig2222}
\end{figure}


Figure~\ref{fig2222} illustrates the evolution of the uncertainty measure 
$J(IT_{2,4}(t))$ for the inactivity times of failed components in eight coherent systems characterized by distinct signature vectors, under a Weibull model with parameters $\alpha = 0.4$ and $\lambda = 1.5$. 
This measure reflects the dynamic uncertainty associated with the residual inactivity of failed components, where larger (less negative) values of $J(IT_{2,4}(t))$ indicate greater predictability and structural stability.

At small values of $t$, the systems exhibit similar trends; however, as time progresses, clear distinctions emerge among their reliability patterns. 
System 3 (green curve) consistently maintains the highest $J(IT_{2,4}(t))$ values, signifying the lowest level of uncertainty and the most robust reliability performance throughout the observation horizon. 
This indicates that the inactivity periods of failed components in System~3 are highly predictable, reflecting its superior structural configuration.

In contrast, System 6 (brown curve) shows the lowest $J(IT_{2,4}(t))$ values, implying the highest uncertainty and the most erratic behavior in the inactivity times. 
Systems~5, 7, and 8 follow closely in the lower range of $J(IT_{2,4}(t))$, suggesting comparably unstable reliability structures. 
Systems~1 and~4 occupy intermediate positions, displaying moderate uncertainty levels, while System~2 ranks just below System~3, indicating relatively high stability and reliability.

Overall, the systems can be ranked from \emph{highest to lowest uncertainty} (i.e., from smallest to largest $J(IT_{2,4}(t))$) as follows:
\[
\text{System 6} > \text{System 8} > \text{System 5} > \text{System 7} \approx \text{System 1} > \text{System 4} > \text{System 2} > \text{System 3}.
\]
This comparative analysis demonstrates that the $J(IT_{2,4}(t))$ measure provides a sensitive and discriminating framework for evaluating system-level uncertainty. 
Systems with smaller $J(IT_{2,4}(t))$ values experience higher unpredictability in component inactivity and thus demand more frequent inspection and preventive maintenance, whereas those with larger values, particularly System~3 exhibit enhanced reliability and structural robustness, allowing for more flexible maintenance strategies and longer operational lifetimes.



%\section{Nonparametric Estimators }
%In this section, we introduce a nonparametric approach to estimating the CRJ of the mixed used system  defined in \eqref{4}. Let us assume we have a sequence of $m$ i.i.d. random variables $Y_r$ for $r=1,...,m$
%with pdf $g(y)$, cdf $G(y)$, and survival function $\bar{G}(y)$. Silverman (2018) proposed a kernel density estimator for the pdf $g(y)$  as follows
%\begin{equation}
%g_m(y)=\frac{1}{mh_m}\sum\limits_{r=1}^{m}K\left(\frac{y-Y_r}{h_m}\right),~~x\in\mathbb{R},
%\end{equation} 
%where $h_m$ is the bandwidth or smoothing parameter and $K(.)$ is the kernel function. The bandwidth sequence
%$h_m$ is chosen such that  $mh_m\rightarrow \infty$ as $m\rightarrow \infty$. This ensures that the kernel density estimator $g_m(y)$ converges to the true PDF $g(y)$ as the sample size increases. The kernel function $K(.)$ is a
%symmetric PDF with finite variance. Some commonly used kernel functions include the normal (Gaussian), Epanechnikov, and tricube kernels. Hereafter, we will use the normal kernel function. Once we have the kernel density estimator $g_m(y)$, we can define nonparametric kernel-based estimators of CRJ of the mixed used system  as follows

%\begin{eqnarray}\label{4}
%\xi J_m(T(n,t))=
%-\frac{1}{2}\int_{G_m(t)}^{1}\frac{{\bar{G}}^2_{V_t,m}(v)}{g_m(G_m^{-1}(v))}dv,
%\end{eqnarray}
%where
%\begin{equation}\label{5}
%{\bar{G}}_{V_t,m}(v)=\sum\limits_{i=1}^{n}p_{i,m}(t){\bar{G}}_{V_{t,i,m}}(v),
%\end{equation}
%and
%\begin{equation}\label{9}
%{\bar{G}}_{V_{t,i,m}}(v)=\sum\limits_{j=0}^{i-1}{n \choose j}\frac{v^{i}(1-v)^{n-i} }{\bar{G}_{i:n,m}(t)},~~ G_m(t)<v<1,
%\end{equation}
%where $p_{i,m}(t)=\frac{s_{i}\bar{G}_{i:n,m}(t)}{\bar{G}^m_{T(n)}(t)}$ for $i=1,\ldots,n$,
%$
%\bar{G}_{i:n,m}(t)=\sum\limits_{j=0}^{i-1}{n \choose j}{G_m(t)}^{j}(1-G_m(t))^{n-j} 
%$, 
%$
%\bar{G}^m_{T(n)}(t)=\sum\limits_{i=1}^{n}s_{i}\bar{G}_{i:n,m}(t)
%$,
 %$G_m(t)=\int_{0}^{t}g_m(y)~dy$, and $\bar{G}_m(t)=\int_{t}^{\infty}g_m(y)~dy$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Non-parametric Estimators for $J(IT_{j,n}(t))$}
In this section, two non-parametric estimators are proposed: a plug-in estimator, and a kernel-smoothed log-odds estimator.  Let
$
R(t) = \frac{G(t)}{\bar{G}(t)} .
$
For a random sample $X_1,\dots,X_N$, let the empirical estimator of the CDF be
\[
\widehat{G}(t) = \frac{1}{N}\sum_{r=1}^N \mathbf{1}\{X_r \le t\},
\qquad
\widehat{\bar{G}}(t)=1-\widehat{G}(t),
\]
%or, in the presence of right censoring, let $\widehat{G}(t)=1-\widehat{S}_{\text{KM}}(t)$ where $\widehat{S}_{\text{KM}}$ is the Kaplan–Meier estimator.
Define the empirical odds
\[
\widehat{R}_{\text{plug}}(t)
= \frac{\widehat{G}(t)}{\max\{1-\widehat{G}(t),\,\varepsilon\}},
\quad \varepsilon>0.
\]
Then replace $R(t)$ by $\widehat{R}_{\text{plug}}(t)$ in \eqref{13} and \eqref{14} to obtain
\[
\widehat{q}_k(t)
= 
\frac{s_k \displaystyle\sum_{l=j}^{k-1} \binom{n}{l}\,\widehat{R}_{\text{plug}}(t)^l}
{\displaystyle\sum_{m=i}^{n}\sum_{l=j}^{m-1}s_m \binom{n}{l}\,\widehat{R}_{\text{plug}}(t)^l},
\]
and 
\[
\widehat{B}_{l,j,k,n}(t)
= \frac{\binom{n}{l}\,\widehat{R}_{\text{plug}}(t)^l}
{\displaystyle\sum_{r=j}^{k-1}\binom{n}{r}\,\widehat{R}_{\text{plug}}(t)^r}.
\]
The plug-in estimator is
\begin{equation}
\widehat{J}_{\text{plug}}(t)
= -\frac{1}{2}\int_0^1 
\left[
\sum_{k=i}^{n}\sum_{l=j}^{k-1}
\widehat{q}_k(t)\,\widehat{B}_{l,j,k,n}(t)\, j\binom{l}{j}\,(1-u)^{j-1}u^{l-j}
\right]^2 du .
\label{eq:J_plugin}
\end{equation}
The integral is evaluated numerically using a trapezoid or Gauss-Legendre rule over a uniform grid $u_m \in [0,1]$.

\begin{algorithm}[H]
\caption{Plug-in Estimator of $J(IT_{j,n}(t))$}
\begin{algorithmic}[1]
\STATE Choose a grid $\{u_m\}_{m=1}^M \subset [0,1]$ and small $\varepsilon>0$.
\STATE Compute $\widehat{G}(t) = \frac{1}{N}\sum 1\{X_i \le t\}$ (or $1-\widehat{S}_{\text{KM}}(t)$ if censored).
\STATE Set $\widehat{R}_{\text{plug}}(t)=\widehat{G}(t)/\max\{1-\widehat{G}(t),\varepsilon\}$.
\FOR{$k=i$ to $n$}
    \STATE $\displaystyle \widehat{q}_k(t) = 
    \frac{s_k \sum_{l=j}^{k-1} \binom{n}{l}\,\widehat{R}_{\text{plug}}(t)^l}
    {\sum_{m=i}^{n}\sum_{l=j}^{m-1}s_m \binom{n}{l}\,\widehat{R}_{\text{plug}}(t)^l}.$
    \FOR{$l=j$ to $k-1$}
        \STATE $\displaystyle \widehat{B}_{l,j,k,n}(t) =
        \frac{\binom{n}{l}\,\widehat{R}_{\text{plug}}(t)^l}
        {\sum_{r=j}^{k-1}\binom{n}{r}\,\widehat{R}_{\text{plug}}(t)^r}.$
    \ENDFOR
\ENDFOR
\STATE For each $u_m$, compute 
$\displaystyle A(u_m;t)=\sum_{k=i}^{n}\sum_{l=j}^{k-1}
\widehat{q}_k(t)\,\widehat{B}_{l,j,k,n}(t)\, j\binom{l}{j}(1-u_m)^{j-1}u_m^{l-j}.$
\STATE Approximate $\displaystyle \widehat{J}_{\text{plug}}(t)
= -\tfrac{1}{2}\sum_{m=1}^{M-1}\frac{u_{m+1}-u_m}{2}
\left[A(u_m;t)^2 + A(u_{m+1};t)^2\right].$
\end{algorithmic}
\end{algorithm}

To reduce the variability of the empirical CDF near 0 or 1, a smooth CDF estimator is used.
Let $K(\cdot)$ be a smooth cumulative kernel (e.g.\ the standard normal CDF) and let $h>0$ be a bandwidth.
Define
\[
\widehat{G}_h(t) = \frac{1}{N}\sum_{r=1}^N K\!\left(\frac{t-X_r}{h}\right),
\]
and 
\[
\widehat{R}_{\text{kerlog}}(t)
= \frac{\widehat{G}_h(t)}{1-\widehat{G}_h(t)}.
\]
Replace $R(t)$ by $\widehat{R}_{\text{kerlog}}(t)$ in
\eqref{13} and \eqref{14} and evaluate
\begin{equation}
\widehat{J}_{\text{kerlog}}(t)
= -\frac{1}{2}\int_0^1 
\left[
\sum_{k=i}^{n}\sum_{l=j}^{k-1}
\widehat{q}_k^{(\text{ker})}(t)\,
\widehat{B}_{l,j,k,n}^{(\text{ker})}(t)\,
j\binom{l}{j}\,(1-u)^{j-1}u^{l-j}
\right]^2 du .
\label{eq:J_kerlog}
\end{equation}

\begin{algorithm}[H]
\caption{Kernel-Logit Estimator of $J(IT_{j,n}(t))$}
\begin{algorithmic}[1]
\STATE Choose kernel $K(\cdot)$ (e.g.\ normal CDF) and bandwidth $h$.
\STATE Compute $\widehat{G}_h(t)=\frac{1}{N}\sum K((t-X_i)/h)$.
\STATE Clip $\widehat{G}_h(t)$ to $[\varepsilon,1-\varepsilon]$ and compute 
$\widehat{R}_{\text{kerlog}}(t)=\widehat{G}_h(t)/(1-\widehat{G}_h(t))$.
\STATE Replace $R(t)$ by $\widehat{R}_{\text{kerlog}}(t)$ in the definitions of 
$q_k(t)$ and $B_{l,j,k,n}(t)$.
\STATE Evaluate the double sum inside the square brackets of \eqref{eq:J_kerlog}
for each $u_m$ and compute the integral numerically.
\STATE Return $\widehat{J}_{\text{kerlog}}(t)$.
\end{algorithmic}
\end{algorithm}



To evaluate the performance of the proposed non-parametric estimators, a simulation study is conducted for the following distributions:
\begin{enumerate}
    \item Weibull distribution with shape parameter $k=0.5$ and scale $\lambda=1.5$:
    \[
    F(x) = 1 - e^{-(x/1.5)^{0.5}}, \quad x>0
    \]
    \item Gamma distribution with shape parameter $k=2$ and scale $\lambda=1.5$:
    \[
    F(x) = \frac{1}{\Gamma(2)} \, \gamma\Big(2, \frac{x}{1.5}\Big), \quad x>0
    \]
    where $\gamma(k,x) = \int_0^x t^{k-1} e^{-t} dt$ is the lower incomplete gamma function.
    \item Exponential distribution with rate parameter $\lambda=1.5$:
    \[
    F(x) = 1 - e^{-1.5 x}, \quad x>0
    \]
    \item Pareto distribution with shape parameter $\alpha=2$ and scale $x_m=1.5$:
    \[
    F(x) = 1 - \Big(\frac{1.5}{x}\Big)^2, \quad x \ge 1.5
    \]
\end{enumerate}
For each distribution, the average bias and root mean square error (RMSE) of the plug-in and kernel-logit estimators of $J(IT_{j,n}(t))$ are computed for various sample sizes $N = 20, 30, 50, 100$ and different system sizes $n$. The smoothing parameter $h$ for the kernel-based estimator is selected using the heuristic formula $h = \lfloor \sqrt{N} + 0.5 \rfloor$, where $\lfloor \cdot \rfloor$ denotes the integer part.  

Each simulation is repeated over 5,000 iterations for $j=n-2$ and $\pmb s =(0,\ldots,0,1)$. The results, presented in Tables~\ref{Weibull},\ref{Gamma},\ref{Exponential}, and \ref{Pareto}  show that as the sample size increases, the RMSE decreases, while the bias tends to increase slightly. These trends are observed consistently across all considered distributions.


\begin{table}[H]
\centering
\small
\caption{Comparison of nonparametric estimators of $J(IT_{j,n}(2))$ for Weibull($k=0.5,\lambda=1.5$)}
\begin{tabular}{cc|cc|cc}
\toprule
$n_{\text{sys}}$ & $N_{\text{sample}}$ &
\multicolumn{2}{c|}{Plug-in} &
\multicolumn{2}{c}{Kerlog} \\
\cline{3-6}
 & & Bias & RMSE & Bias & RMSE \\
\midrule
5  & 20  & -0.00086 & 0.02064 & -0.00149 & 0.01987 \\
5  & 30  & -0.00378 & 0.01906 & -0.00404 & 0.01832 \\
5  & 50  & -0.00312 & 0.01643 & -0.00324 & 0.01573 \\
5  & 100 & -0.00052 & 0.01057 & -0.00094 & 0.01057 \\
7  & 20  & -0.00549 & 0.05835 & -0.00694 & 0.05537 \\
7  & 30  &  0.00090 & 0.05749 &  0.00070 & 0.05612 \\
7  & 50  & -0.00042 & 0.04166 & -0.00144 & 0.03971 \\
7  & 100 &  0.00070 & 0.03178 & -0.00148 & 0.03137 \\
10 & 20  &  0.00580 & 0.11947 &  0.00308 & 0.11879 \\
10 & 30  &  0.00179 & 0.10592 & -0.00181 & 0.10083 \\
10 & 50  &  0.00084 & 0.07872 &  0.00017 & 0.07632 \\
10 & 100 &  0.00549 & 0.05839 &  0.00024 & 0.05634 \\
15 & 20  &  0.02727 & 0.20715 &  0.02036 & 0.19422 \\
15 & 30  &  0.01038 & 0.13987 &  0.00339 & 0.13124 \\
15 & 50  &  0.00893 & 0.12972 &  0.00330 & 0.12327 \\
15 & 100 &  0.02127 & 0.09456 &  0.01565 & 0.09012 \\
\bottomrule
\end{tabular}
\label{Weibull}
\end{table}


For the Weibull distribution with shape parameter $k=0.5$ and scale $\lambda=1.5$, both the plug-in and kernel-logit estimators demonstrate satisfactory performance across all sample sizes. The bias of both estimators remains close to zero, indicating accurate estimation of $J(IT_{j,n}(t))$. However, the kernel-logit estimator consistently exhibits slightly lower RMSE values, particularly for smaller sample sizes, suggesting a more stable estimation in the presence of limited data. This trend highlights the advantage of the kernel-based smoothing in reducing variability without introducing significant bias (see Table~\ref{Weibull}).







\begin{table}[H]
\centering
\small
\caption{Comparison of nonparametric estimators of $J(IT_{j,n}(2))$ for Gamma($k=2,\lambda=1.5$)}
\begin{tabular}{cc|cc|cc}
\toprule
$n_{\text{sys}}$ & $N_{\text{sample}}$ &
\multicolumn{2}{c|}{Plug-in} &
\multicolumn{2}{c}{Kerlog} \\
\cline{3-6}
 & & Bias & RMSE & Bias & RMSE \\
\midrule
5  & 20  & -0.00220 & 0.03397 & -0.00151 & 0.02964 \\
5  & 30  &  0.00247 & 0.02822 &  0.00177 & 0.02434 \\
5  & 50  & -0.00533 & 0.02195 & -0.00499 & 0.02041 \\
5  & 100 & -0.00145 & 0.01497 & -0.00197 & 0.01395 \\
7  & 20  & -0.00785 & 0.05464 & -0.00816 & 0.04921 \\
7  & 30  & -0.00160 & 0.04481 & -0.00162 & 0.04078 \\
7  & 50  &  0.00466 & 0.03949 &  0.00267 & 0.03537 \\
7  & 100 & -0.00240 & 0.02570 & -0.00216 & 0.02382 \\
10 & 20  &  0.00056 & 0.07841 & -0.00317 & 0.06930 \\
10 & 30  &  0.00286 & 0.06390 &  0.00318 & 0.05797 \\
10 & 50  &  0.00789 & 0.05495 &  0.00666 & 0.05038 \\
10 & 100 &  0.00294 & 0.03796 &  0.00098 & 0.03471 \\
15 & 20  &  0.02552 & 0.11258 &  0.02194 & 0.10375 \\
15 & 30  &  0.00587 & 0.08562 &  0.00041 & 0.07936 \\
15 & 50  &  0.00493 & 0.07530 &  0.00348 & 0.07064 \\
15 & 100 &  0.00369 & 0.04620 &  0.00233 & 0.04101 \\
\bottomrule
\end{tabular}
\label{Gamma}
\end{table}

For the Gamma distribution with shape $k=2$ and scale $\lambda=1.5$, a similar pattern is observed. While the biases of both estimators are generally small, the kernel-logit estimator tends to provide a slightly lower RMSE for most sample sizes, especially when $N$ is small or moderate. These results indicate that kernel smoothing effectively enhances estimator performance by controlling random fluctuations in the empirical CDF, leading to more reliable estimates of $J(IT_{j,n}(t))$ (see Table~\ref{Gamma}).

\begin{table}[H]
\centering
\small
\caption{Comparison of nonparametric estimators of $J(IT_{j,n}(2))$ for Exponential($\lambda = 1.5$)}
\begin{tabular}{cc|cc|cc}
\toprule
$n_{\text{sys}}$ & $N_{\text{sample}}$ &
\multicolumn{2}{c|}{Plug-in} &
\multicolumn{2}{c}{Kerlog} \\
\cline{3-6}
 & & Bias & RMSE & Bias & RMSE \\
\midrule
5  & 20  & -0.00361 & 0.00853 & -0.00195 & 0.00703 \\
5  & 30  & -0.00087 & 0.00599 &  0.00028 & 0.00485 \\
5  & 50  & -0.00120 & 0.00504 &  0.00004 & 0.00413 \\
5  & 100 & -0.00033 & 0.00360 &  0.00041 & 0.00311 \\
7  & 20  & -0.00780 & 0.01315 & -0.00715 & 0.01236 \\
7  & 30  & -0.00682 & 0.01138 & -0.00586 & 0.01089 \\
7  & 50  & -0.00366 & 0.00637 & -0.00257 & 0.00493 \\
7  & 100 & -0.00157 & 0.00292 & -0.00180 & 0.00342 \\
10 & 20  & -0.02116 & 0.04937 & -0.01953 & 0.04429 \\
10 & 30  & -0.01048 & 0.03667 & -0.01229 & 0.03827 \\
10 & 50  & -0.00527 & 0.02060 & -0.00737 & 0.02202 \\
10 & 100 & -0.00582 & 0.01827 & -0.00951 & 0.02004 \\
15 & 20  & -0.03681 & 0.13873 & -0.04245 & 0.13533 \\
15 & 30  & -0.02090 & 0.11396 & -0.02694 & 0.10861 \\
15 & 50  & -0.00216 & 0.07754 & -0.01329 & 0.07142 \\
15 & 100 & -0.01809 & 0.07900 & -0.02493 & 0.07326 \\
\bottomrule
\end{tabular}
\label{Exponential}
\end{table}



In the case of the Exponential distribution with rate $\lambda=1.5$, the performance of both estimators is again comparable, with bias values near zero. The kernel-logit estimator achieves marginally lower RMSEs across most sample sizes, demonstrating its ability to provide slightly more precise estimates in small samples. The differences between the two estimators diminish as the sample size increases, reflecting the convergence properties of the plug-in approach with large datasets (see Table~\ref{Exponential}).

\begin{table}[H]
\centering
\small
\caption{Comparison of nonparametric estimators of $J(IT_{j,n}(2))$ for Pareto($\alpha = 2$, $x_m = 1.5$)}
\begin{tabular}{cc|cc|cc}
\toprule
$n_{\text{sys}}$ & $N_{\text{sample}}$ &
\multicolumn{2}{c|}{Plug-in} &
\multicolumn{2}{c}{Kerlog} \\
\cline{3-6}
 & & Bias & RMSE & Bias & RMSE \\
\midrule
5  & 20  & -0.00184 & 0.03402 & -0.01142 & 0.02780 \\
5  & 30  &  0.00013 & 0.02957 & -0.00785 & 0.02304 \\
5  & 50  & -0.00120 & 0.02188 & -0.01108 & 0.02004 \\
5  & 100 & -0.00188 & 0.01748 & -0.01132 & 0.01764 \\
7  & 20  &  0.00702 & 0.05024 & -0.01182 & 0.04027 \\
7  & 30  &  0.00339 & 0.05210 & -0.01914 & 0.04227 \\
7  & 50  &  0.00025 & 0.03981 & -0.01882 & 0.03555 \\
7  & 100 &  0.00019 & 0.02738 & -0.01871 & 0.02780 \\
10 & 20  &  0.00629 & 0.09182 & -0.02061 & 0.07030 \\
10 & 30  &  0.00539 & 0.07527 & -0.01900 & 0.05961 \\
10 & 50  &  0.00487 & 0.06270 & -0.02288 & 0.04774 \\
10 & 100 & -0.00137 & 0.03907 & -0.02781 & 0.04068 \\
15 & 20  &  0.01109 & 0.12042 & -0.02524 & 0.08370 \\
15 & 30  &  0.01962 & 0.09127 & -0.02305 & 0.05936 \\
15 & 50  &  0.00560 & 0.07622 & -0.02656 & 0.06007 \\
15 & 100 & -0.00527 & 0.04510 & -0.03656 & 0.04816 \\
\bottomrule
\end{tabular}
\label{Pareto}
\end{table}



For the Pareto distribution with shape $\alpha=2$ and scale $x_m=1.5$, the behavior differs slightly. While both estimators remain reasonably accurate, the plug-in estimator often exhibits a lower RMSE than the kernel-logit estimator for larger sample sizes, despite comparable bias levels. This suggests that for heavy-tailed distributions, the plug-in approach may offer slightly more stable performance, whereas kernel smoothing may introduce additional variability in the tails. Overall, the simulation results demonstrate that both estimators perform well, with kernel-logit generally preferable for light- to moderate-tailed distributions and plug-in slightly advantageous for heavy-tailed distributions (see Table~\ref{Pareto}).
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% GENERAL THEOREMS FOR J(IT_{j,n}(t)) -- Plug-in, Kernel-Logit, Affine
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\subsection{Consistency and equivariance results for $J(IT_{j,n}(t))$}

We study the functional
\[
J\big(IT_{j,n}(t);R\big)
=-\frac{1}{2}\int_{0}^{1}
\Bigg( \sum_{k=i}^{n}\sum_{l=j}^{k-1} q_k(R)\,B_{l,j,k,n}(R)\; j\binom{l}{j}(1-u)^{j-1}u^{\,l-j}
\Bigg)^2 du,
\]
where for a given odds function $R>0$ the coefficients are defined (whenever denominators are non-zero) by
\[
q_k(R)=\frac{s_k\displaystyle\sum_{l=j}^{k-1}\binom{n}{l}R^l}
{\displaystyle\sum_{m=i}^{n} s_m\sum_{l=j}^{m-1}\binom{n}{l}R^l},
\qquad
B_{l,j,k,n}(R)=\frac{\binom{n}{l}R^l}{\displaystyle\sum_{r=j}^{k-1}\binom{n}{r}R^r}.
\]
We assume the weight vector $s=(s_1,\dots,s_n)$ is fixed and that for the evaluation point $t$ the true CDF satisfies $0<G(t)<1$. Standard measurability (continuity)  conditions for the finite sums below are then satisfied.

% ------------------------------
% ------------------------------

\begin{theorem}[Strong consistency of the plug-in estimator]
Let $X_1,\dots,X_N$ be IID  with CDF $G$. Fix $n,i,j$ as above and a point $t$ with $0<G(t)<1$. Define
$
\widehat G_N(t)=\frac{1}{N}\sum_{m=1}^N \mathbf 1\{X_m\le t\},$
 $\widehat R_N(t)=\frac{\widehat G_N(t)}{1-\widehat G_N(t)},
$
and the plug--in estimator $\widehat J_{N,\mathrm{plug}}(t)=J(IT_{j,n}(t);\widehat R_N(t))$.
Then
\begin{equation}
\widehat J_{N,\mathrm{plug}}(t)\xrightarrow{\ \mathrm{a.s.}\ } J(IT_{j,n}(t);R(t))
\quad\text{as }N\to\infty,
\end{equation}
where $R(t)=G(t)/(1-G(t))$.
\end{theorem}

{\bf{Proof.}}
We first note that by the Strong Law of Large Numbers (SLLN) applied to the indicator function at the 
fixed point $t$,             we have $\widehat G_N(t)\to G(t)$ almost surely (see e.g.\ Billingsley (1995) or Feller for the classical SLLN). Because $0<G(t)<1$ and $\widehat G_N(t)\in[0,1]$ almost surely, we may apply the continuous mapping theorem (see van der Vaart (1998), Theorem 1.3.6) to the continuous function $x\mapsto x/(1-x)$ on a neighborhood of $G(t)$; this gives $\widehat R_N(t)\to R(t)$ almost surely.

Next, consider the inner summand
\[
A(u;R)=\sum_{k=i}^{n}\sum_{l=j}^{k-1} q_k(R)\,B_{l,j,k,n}(R)\; j\binom{l}{j}(1-u)^{j-1}u^{\,l-j},
\]
viewed as a function of $(u,R)\in[0,1]\times(0,\infty)$. Each coefficient $q_k(R)$ and $B_{l,j,k,n}(R)$ is a ratio of finite sums of continuous functions $R\mapsto R^l$ with strictly positive denominators for $R>0$ (given the index ranges and non-negative binomial coefficients). Thus $q_k$ and $B_{l,j,k,n}$ are continuous in $R$ on $(0,\infty)$, and therefore $A(u;R)$ is continuous in $(u,R)$. Since $[0,1]$ is compact and the finite summation yields bounded integrands for $R$ in a neighborhood of $R(t)$, the map $R\mapsto \int_0^1 A(u;R)^2 du$ is continuous by an application of the Dominated Convergence Theorem (DCT) or uniform continuity on compacta; see Folland (1999) or Royden for DCT. Concretely, for any sequence $R_n\to R$ we have $A(\cdot,R_n)^2\to A(\cdot,R)^2$ pointwise and the integrands are uniformly bounded on $[0,1]$ by a finite constant determined by combinatorial coefficients and a neighborhood bound on $R$, so DCT yields convergence of integrals.

Combining the two convergence results: $\widehat R_N(t)\to R(t)$ a.s. and continuity of $R\mapsto J(IT_{j,n}(t);R)$ at $R(t)$, we obtain
\[
\widehat J_{N,\mathrm{plug}}(t)=J(IT_{j,n}(t);\widehat R_N(t))\xrightarrow{\mathrm{a.s.}} J(IT_{j,n}(t);R(t)),
\]
as required. 
$\hfill \Box$

% ------------------------------
% ------------------------------
\begin{theorem}[Consistency of the kernel--logit estimator]
Under the setting of Theorem 1, assume additionally that $\mathbb E|X_1|^p<\infty$ for some $p>1$. Let $h_N>0$ be a bandwidth sequence with $h_N\downarrow 0$ and $N h_N\to\infty$. Define the smoothed estimator
\[
\widehat G_{N,h}(t)=\frac{1}{N}\sum_{m=1}^N \Phi\!\Big(\frac{t-X_m}{h_N}\Big),
\qquad
\widehat R_{N,h}(t)=\frac{\widehat G_{N,h}(t)}{1-\widehat G_{N,h}(t)},
\]
and set $\widehat J_{N,\mathrm{kerlog}}(t)=J(IT_{j,n}(t);\widehat R_{N,h}(t))$. Then for fixed $t$,
\[
\widehat J_{N,\mathrm{kerlog}}(t)\xrightarrow{\;p\;} J(IT_{j,n}(t);R(t))\quad (N\to\infty).
\]
If uniform a.s.\ convergence of $\widehat G_{N,h}$ holds under stronger regularity, the convergence may be strengthened to almost sure.
\end{theorem}

{\bf{Proof.}}
Kernel smoothing theory for distribution function estimators (see the discussion in Silverman (1986) and Wand \& Jones (1995)) yields that under mild regularity—most importantly $h_N\to 0$ and $N h_N\to\infty$—the smoothed estimator $\widehat G_{N,h}(t)$ is consistent for $G(t)$ in probability for each fixed $t$; the moment condition $\mathbb E|X_1|^p<\infty$ with $p>1$ ensures control of boundary/bias terms and integrability required for the usual bias–variance analysis. Thus $\widehat G_{N,h}(t)\xrightarrow{p}G(t)$.

Applying the continuous mapping theorem (van der Vaart (1998)) to the continuous map $x\mapsto x/(1-x)$ gives $\widehat R_{N,h}(t)\xrightarrow{p}R(t)$. The remaining step is identical in spirit to the argument in Theorem 1: the functional $R\mapsto J(IT_{j,n}(t);R)$ is continuous at $R(t)$ because its integrand is jointly continuous in $(u,R)$ and dominated on $[0,1]$ by an integrable bound for $R$ near $R(t)$. Therefore, by another application of the continuous mapping theorem, we obtain $\widehat J_{N,\mathrm{kerlog}}(t)=J(\widehat R_{N,h}(t))\xrightarrow{p}J(R(t))$, as claimed.
$\hfill \Box$

% ------------------------------
% ------------------------------


\begin{theorem}[Affine equivariance and moment identities]
Let $Y_i=cX_i+b$ with $c>0$, $b\in\mathbb R$. Denote by $G_X$ and $G_Y$ the CDFs of $X$ and $Y$, by $R_X,R_Y$ their odds, and by $\widehat J_{m,X},\widehat J_{m,Y}$ any plug-in–type estimators obtained by substituting an empirical or smoothed CDF into the functional $J$ (this includes plug-in, kernel--logit, and leave-one-out jackknife versions). Then for each fixed $t$,
\[
J_Y(t)=J_X\Big(\frac{t-b}{c}\Big),
\qquad
\widehat J_{m,Y}(t)=\widehat J_{m,X}\Big(\frac{t-b}{c}\Big),
\]
and consequently expectations, variances and RMSEs satisfy the corresponding identities:
\[
\mathbb E[\widehat J_{m,Y}(t)] = \mathbb E\!\Big[\widehat J_{m,X}\Big(\frac{t-b}{c}\Big)\Big],
\]
\[
\operatorname{Var}(\widehat J_{m,Y}(t)) = \operatorname{Var}\!\Big(\widehat J_{m,X}\Big(\frac{t-b}{c}\Big)\Big),
\quad
\operatorname{RMSE}(\widehat J_{m,Y}(t)) = \operatorname{RMSE}\!\Big(\widehat J_{m,X}\Big(\frac{t-b}{c}\Big)\Big).
\]
\end{theorem}

{\bf{Proof.}}
The identity $G_Y(t)=P(Y\le t)=P(X\le (t-b)/c)=G_X((t-b)/c)$ is immediate from the definition $Y=cX+b$. Consequently $R_Y(t)=R_X((t-b)/c)$ for each $t$. Because the functional $J$ depends on the distribution only through the odds function $R$, it follows that $J_Y(t)=J_X((t-b)/c)$.

For estimators formed by substitution, plug-in versions satisfy exactly $\widehat G_{N,Y}(t)=\widehat G_{N,X}((t-b)/c)$ (empirical indicators transform accordingly), and the same relation holds for smoothed kernel versions after the natural bandwidth rescaling $h_Y=c\,h_X$ so that smoothing on the $Y$-scale corresponds to smoothing on the $X$-scale. Therefore $\widehat R_{Y}(\cdot)=\widehat R_{X}((\cdot-b)/c)$ and substituting into $J$ yields $\widehat J_{m,Y}(t)=\widehat J_{m,X}((t-b)/c)$. The equalities of expectations, variances and RMSE then follow directly by taking expectation/variance of both sides (no asymptotic argument required). 
$\hfill \Box$










\section{Image Processing Application Using the $J(IT_{j,n}(t))$ Measure}

In this section, we demonstrate how the proposed estimator of 
$J(IT_{j,n}(t))$ can be applied to quantify structural uncertainty in digital images. 
Following the methodology of Toomaj and Atabay (2022) and Toomaj (2023), we begin with a normalized grayscale 
image denoted by $X$. \ph{In our experiment, $X$ corresponds to a grayscale photograph of a cat. 
The image used in the image processing experiment is a publicly available photograph obtained from Wikimedia Commons (https://upload.wikimedia.org/wikipedia/commons/3/3a/Cat03.jpg). The original image is licensed for public use under the terms stated on its source page. }
We generate three modified versions of the image through deterministic intensity transformations:
\[
Y = 0.9X + 0.1, 
\qquad 
Z = X^{2},
\qquad
W = \sqrt{X}.
\]
The transformation $Y$ introduces a mild linear modification while largely preserving the contrast 
and structural information of the original image $X$. In contrast, $Z$ applies a convex power 
transformation that amplifies bright intensities, whereas $W$, being a concave transformation, stretches 
low-intensity regions and generally leads to a loss of contrast and detail.
 


\begin{figure}[H] 
\centering
\includegraphics[scale=0.6]{JIT_Images_XYZW_HD.png} 
\vspace*{-0.5cm}\caption{ \footnotesize{
Sample picture of a cat with their adjustments
}}
\label{fig2222}
\end{figure}


\begin{figure}[H] 
\centering
\vspace*{-0.6cm}
\includegraphics[scale=0.9]{Histogram.eps} 
\vspace*{-0.5cm}\caption{ \footnotesize{
The histograms for pictures $X$, $Y$, $Z$ and $W$.
  }}\label{fig11}
\end{figure}

For each image $(X, Y, Z, W)$, we extract a random sample of pixel intensities and compute the 
plug-in and Kerlog estimators of $J(IT_{j,n}(t))$ on the grid $t \in [0, 0.8]$. The two sets of system parameters considered in this study are
\[
n = 10, \quad i = 10, \quad j = 8, \quad \boldsymbol{s} = (0,\ldots,0,1),
\]
and
\[
n = 4, \quad i = 2, \quad j = 2, \quad \boldsymbol{s} = \left(0, \frac{1}{6}, \frac{7}{12}, \frac{1}{4}\right).
\]
The corresponding estimators were then computed and their results are presented in Figures \ref{fig:XYZ_example1} and \ref{fig:XYZ_example2}.
The resulting measure $J(IT_{j,n}(t))$ contains information about the inactivity time distribution 
of a system constructed from the pixel intensity sample. Therefore, differences between the 
estimated curves provide a quantitative means to distinguish between structural changes introduced by 
different intensity transformations.


\begin{figure}[H]
\centering
% --- دو تصویر بالا ---
\includegraphics[scale=0.5]{plug-paralleln10i10.eps}
\hspace{0.4cm}
\includegraphics[scale=0.5]{kerlogparallel-n1i10.eps}

\\[0.3cm] % فاصله عمودی بین ردیف بالا و پایین

% --- تصویر پایین در وسط ---

\vspace*{-0.3cm}
\caption{\footnotesize{
The values of estimators $\widehat{J}_{\text{kerlog}}(t)$ and $\widehat{J}_{\text{plug}}(t)$.
}}
\label{fig:XYZ_example1}
\end{figure}



\begin{figure}[H]
\centering
% --- دو تصویر بالا ---
\includegraphics[scale=0.5]{plug-n4i4j4.eps}
\hspace{0.4cm}
\includegraphics[scale=0.5]{kerlog-n4i2j2.eps}

\\[0.3cm] % فاصله عمودی بین ردیف بالا و پایین

% --- تصویر پایین در وسط -
\vspace*{-0.3cm}
\caption{\footnotesize{
The values of estimators $\widehat{J}_{\text{kerlog}}(t)$ and $\widehat{J}_{\text{plug}}(t)$.
}}
\label{fig:XYZ_example2}
\end{figure}
The estimated curves reveal a clear and consistent pattern. The curve associated with $Y$ lies extremely 
close to that of the original image $X$, reflecting the fact that the transformation $Y = 0.9X + 0.1$ 
preserves the relative ordering of pixel intensities and leaves the overall distribution nearly unchanged.

The curve corresponding to $Z = X^{2}$ shows a moderate deviation from $X$, which is in line with the 
behavior of a convex power transformation that enhances bright regions more heavily than dark ones. 
Nonetheless, the structural relationship between $Z$ and $X$ remains relatively strong.

In contrast, the transformation $W = \sqrt{X}$ induces a substantial shift, and the estimated 
$J(IT_{j,n}(t))$ curve diverges noticeably from those of $X$, $Y$, and $Z$ for almost all $t$. 
This is expected, as the concave power transformation significantly flattens the intensity distribution 
and weakens structural dependencies.


These results show that the proposed estimator is sensitive to photometric distortions and captures 
distributional differences between images in a meaningful way. Transformations that preserve the 
monotonic structure of intensities (such as $Y$) produce curves close to the original image, while 
non-linear modifications (particularly $W$) generate clearly distinguishable deviation patterns.

Thus, the measure $J(IT_{j,n}(t))$ serves as a robust tool for assessing structural alterations in 
digital images and provides a principled, distribution-based method for quantifying image uncertainty.





\section{Concluding Remarks}

This paper provides a comprehensive investigation into the extropy associated with the inactivity times of failed components in coherent systems. By employing the system signature framework, we derived explicit representations for the extropy of inactivity durations and established several stochastic bounds and comparative results that reveal how system structure and configuration influence the informational characteristics of such hidden failure behavior. These theoretical developments offer a novel information-theoretic perspective on reliability analysis, complementing classical hazard-based and lifetime-based approaches that frequently overlook the latent contribution of failed components that persist within an otherwise functioning system.

On the methodological side, we introduced two non-parametric estimators for the extropy of inactivity times and examined their performance using simulated datasets under a variety of lifetime models. The numerical results demonstrate that both estimators provide reliable approximations, with the kernel-logit estimator showing superior performance for light- and medium-tailed distributions, whereas the plug-in estimator exhibits greater stability for heavy-tailed cases. These findings highlight the practical relevance of extropy-based measures in settings where the distributional behavior of inactivity times is complex or partially obscured.

Finally, the applicability of the proposed framework was illustrated through an image processing example, where extropy successfully captured subtle structural and contrast-related features under different intensity transformations. This demonstrates that the informational insights derived from extropy extend beyond classical reliability contexts and can be utilized in broader data-analytic and signal-processing environments. Overall, the results presented in this work emphasize the significance of inactivity times in understanding the hidden dynamics of coherent systems and underscore the potential of extropy as a powerful tool for both theoretical investigation and practical analysis.




\section*{Declarations} 
{\bf Ethics approval and consent to participate}\\
 This study does not involve human participants or animals. Therefore, ethical approval is not applicable.\\
  {\bf Competing interests }\\
  The authors declare no competing interests.\\
{\bf Funding information}\\
 No funding was received for conducting this study. \\
 {\bf Data Availability }\\
The image used in the image-processing experiment is a publicly available photograph obtained from Wikimedia Commons ( \texttt{https://upload.wikimedia.org/wikipedia/commons/3/3a/Cat03.jpg}). 
The original image is licensed for public use under the terms stated on its source page. The pixel–intensity samples employed in the numerical analysis, as well as the R code (file name: \texttt{ CatPic.R}) used to generate the transformed images ( file name: \texttt{Image\_XYZW\_Pixel\_Samples.xls}
) and compute the estimators, have been uploaded as supplementary materials on the manuscript’s submission page. No additional datasets were generated or analysed in the current study.\\
 
 
\section*{ Author Contributions}
Zohreh Pakdaman and Reza Alizadeh Noughabi jointly developed the main theoretical framework, derived the mathematical results, and performed the analytical proofs. Both authors contributed equally to the writing of the manuscript, preparation of figures and numerical illustrations, and interpretation of the results. All authors reviewed and approved the final version of the manuscript. Both authors contributed equally to this work. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Acknowledgements}
%The authors sincerely thank the anonymous reviewers for their constructive comments and valuable suggestions, which helped improve the quality and clarity of this paper.
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