A variational derivation of the field equations of an action-dependent Einstein-Hilbert Lagrangian

1.   Introduction

The Farlie-Gumbel-Morgenstern (FGM) family of bivariate distributions has been widely utilized in practical applications. This family is defined by the given marginal distribution functions (DFs) $F_Y(y)$ and $F_Z(z)$ for the random variables (RVs) $Y$ and $Z,$ respectively, along with a parameter $\alpha,$ resulting in the bivariate DF given by

with the corresponding probability density function (PDF)

Here $f_{Y}(y)$ and $f_{Z} (z)$ are the marginals of $f_{Y, Z}(y, z),$ and $\alpha$ is referred to as the association parameter. The RVs $Y$ and $Z$ are treated as independent at $\alpha = 0.$ The initial introduction of this model was done by ^[1] and further examined by ^[2] for exponential marginals. The overall structure described in Eq (1.1) is credited to ^[3,4]. The parameter $\alpha$ must lie in the range $[-1, 1]$ to ensure that the function $F_{Y, Z}(y, z)$ remains a valid DF. The Pearson correlation coefficient $\rho$ between $Y$ and $Z$ cannot exceed $\frac{1}{3}.$ An aspect of the behavior of the bivariate PDF with the associated parameter $\alpha$ for the FGM association (i.e., the FGM family with uniform margins) is depicted in Figure 1.

Reference ^[5] proposed the concept of the GOSs as a comprehensive framework for RVs arranged in ascending order. The subclass $m-$GOSs of GOSs encompasses the most prominent models of ordered RVs. These models include OSs, lower and upper record values, $k-$records, sequential OSs, and progressive type Ⅱ censoring with a constant scheme. We exclusively consider the $m-$GOSs model. Consider $F_Y(y) = P(Y\leq y)$ as an arbitrary continuous DF with the PDF $f_Y(y).$ Then the RVs $Y(1, n, m, k)\leq Y(2, n, m, k)\leq...\leq Y(n, n, m, k)$ ($k > 0, \; m\geq-1$) are said to be $m-$GOSs if their joint PDF (JPDF) is given by

where $F_Y^{-1}(1)\geq y_n\geq...\geq y_1\geq F_Y^{-1}(0)$ and $\gamma_j = k+(n-j)(m+1) > 0, j = 1, 2, ..., n$ (note that $\gamma_n = k$). The marginal PDF of $r$th $m-$GOS, $1\leq r\leq n,$ is given by (cf. ^[5])

where $C_{r-1} = \prod \limits _{i = 1}^{r}\gamma_i, r = 1, 2, ..., n, \; g_{m}(y) = h_{m}(y)-h_{m}(0), \; y\in [0, 1),$ and $h_{m}(y) = \frac{-1}{(m+1)}(1-y)^{m+1},$ if $m\neq-1,$ while $h_{-1}(y) = -\log (1-y).$

The concomitant subject has recently gained traction again because of its applicability to prediction problems and selection processes. The concept of the concomitants of OSs may be traced back to ^[6], while ^[7] further developed the broad theory of concomitants of OSs. Reference ^[6] provided an exceptional examination of the concomitants of OSs. Compared to the concomitants of OSs, the concomitants of GOSs have not been as fully studied. Numerous researchers have studied this topic, including^[8,9,10].

For the FGM family defined by (1.1) and (1.2), the PDF, DF, and survival function (SF) of the concomitant of the $r$th $m$-GOS are given by ^[9], respectively, as

and

where $C^{*}_{(r, n, m, k)} = \frac{2\prod \limits _{j = 1}^{r}\gamma_{j}}{\prod \limits _{i = 1}^{r}(\gamma_{i}+1)}-1$ and $\overline{F}_Z(z) = 1-F_Z(z).$

Reference ^[9] conducted a study on the concomitants of $m$-GOSs in the FGM family. Reference^[11] derived certain characteristics of the concomitants of $m$-GOSs from the bivariate Rayleigh distribution of FGM type. References ^[12,13] examined various information measures in concomitants of $m$-GOSs under iterated FGM and Huang-Kotz FGM, respectively. References ^[14,15,16] examined certain characteristics of concomitants of $m$-GOSs from the bivariate Cambanis family.

Shannon entropy, established by ^[17], is one of the most extensively used measures of uncertainty. It is used to quantify the amount of uncertainty involved in an RV and has multiple uses in various fields. Let $Y$ be a non-negative RV having PDF $f_Y(y).$ ^[17] defined the entropy of $Y$ by

An entropic expression with an index $\theta$ was first presented by ^[18] and results in non-extensive statistics. The so-called non-extensive statistical mechanics, which generalize Boltzmann-Gibbs theory, is based on Tsallis entropy. Applications of Tsallis statistics can be found in a wide range of fields, including physics, chemistry, biology, medicine, economics, and geophysics. According to ^[18], the non-additive generalization of Shannon's entropy of order $\theta$ is called Tsallis entropy. This measure is crucial in the uncertainty assessments of an RV $Y,$ defined as

where $\theta\neq1, \theta > 0.$ Tsallis entropy approaches Shannon entropy as $\theta\longrightarrow 1.$

Tsallis statistics, rooted in the generalized entropy framework, are particularly valuable for systems with long-range interactions, non-equilibrium dynamics, and fractal-like structures. These properties make them highly applicable to fields dealing with complex and noisy data, such as seismic inversion and magnetic resonance imaging evaluations. For example, seismic inversion involves extracting subsurface geological properties (e.g., porosity, fluid content, rock type) from seismic reflection data. These data sets are typically noisy, incomplete, and exhibit long-range spatial correlations, making them ideal candidates for Tsallis statistical approaches. In seismic inversion and MRI evaluations, Tsallis entropy and statistics provide enhanced tools for dealing with noisy, complex, and non-linear data. They enable better noise filtering, improved stability in inversion algorithms, and deeper insights into fractal-like patterns and tissue textures, ultimately leading to more accurate geological modeling and medical diagnoses. For additional information regarding Tsallis entropy, we suggest reading ^[19,20]. Numerous generalizations of Shannon entropy have been formulated, rendering these entropies responsive to various types of probability distributions by the incorporation of several additional factors.

Reference ^[21] presented a novel measure of Shannon entropy known as cumulative residual entropy (CRE), which utilizes the SF rather than the PDF. The CRE is regarded as more stable and mathematically robust due to its more consistent SF compared to the PDF. Moreover, DFs exist even when PDFs do not exist (e.g., Govindarajulu, power-Pareto, and generalized lambda distributions). The CRE measure based on the SF $\overline F_Y(y)$ is defined as

The cumulative residual Tsallis entropy (CRTE) of order $\theta$ is a helpful generalization of Shannon entropy that can be considered an alternate dispersion measure and has demonstrated promising results in several applications. Reference ^[22] devised the measure CRTE, which is defined by

When $\theta\longrightarrow 1$, CRTE approaches CRE.

Reference ^[23] unveiled an alternative measure for CRTE, which is defined by

Unlike CRTE, the measure defined by (1.5) has some additional features and has simple relationships with other important information and reliability measures. Reference ^[24] established a parallel notion of CRE, the cumulative entropy, which is useful for measuring information when uncertainty is associated with the past. Motivated from (1.5), reference ^[25] introduced cumulative past Tsallis entropy (CPTE) which is defined as

Assigning appropriate weights is crucial for accurately reflecting the relative relevance or importance of different observations or events in a data set. Weighting methodologies ensure that observations or events in a data set are appropriately prioritized based on their relevance or importance. Key approaches include:

1) Statistical weighting: based on variance (lower variance = higher weight) or frequency (higher frequency = higher weight).

2) Entropy-based weighting: higher entropy events get higher weights due to their greater uncertainty.

3) Expert knowledge-based weighting: relies on domain experts' insights.

4) Information gain-based weighting: events providing more information gain are assigned higher weights.

5) Distance-based weighting: observations closer to a reference point have higher weights (e.g., inverse distance weighting).

Each methodology suits different scenarios and data set characteristics, and the choice depends on the objective and data context. Reference ^[26] defined weighted CPTE (WCPTE) of order $\theta$ as

It was shown that the WCPTE can be used as a risk measure. Moreover, ^[27] proposed weighted CRTE (WCRTE) defined as

For a non-negative continuous RV $Y,$ ^[26] defined a new measure called "alternative weighted cumulative residual Tsallis entropy" (denoted as AWCRTE) by

Despite the extensive use of Tsallis entropy in fields such as medical imaging (e.g., see ^[28,29]), there remains a need for further exploration of how Tsallis entropy can be applied to more complex, dynamic systems$-$especially when considering weighted measures that reflect cumulative effects over time.

This paper introduces the concepts of WCRTE and WCPTE, along with their dynamic versions, derived from $m$-GOSs in the context of the FGM family. The motivation for this work stems from the desire to enhance the versatility and applicability of Tsallis entropy in systems where dynamic and weighted measures are essential, such as in medical imaging, environmental studies, and artificial intelligence. The novel entropy measures introduced here offer new approaches for analyzing non-extensive and complex systems that cannot be adequately described by traditional entropy measures, such as Shannon entropy. The integration of these new tools with existing models opens up new possibilities for understanding and predicting behaviors in dynamic systems.

This is how the remainder of the paper is structured. Section 2 discusses some cases and provides the WCRTE for the concomitant $Z_{[r, n, m, k]}$ based on the FGM family of bivariate distributions. In the same section, we suggest a dynamic version of AWCRTE and its characteristics for $Z_{[r, n, m, k]}$ based on this family. In Section 3, the WCPTE and its dynamic form are examined. In Section 4, we offer the empirical AWCRTE and WCPTE. In Section 5, two real data sets are analyzed for illustration. In Section 6, the paper outlines the study's essential findings and their ramifications.

2.   Weighted cumulative residual Tsallis entropy of order $\theta$

Theorem 2.1. The WCRTE of concomitant $Z_{[r, n, m, k]}$ of the $r$th $m-$GOS based on the FGM family is given by

where ${\cal{N}}(z) = \infty,$ if $z$ is a non-integer, ${\cal{N}}(z) = z,$ if $z$ is an integer, $U$ is the uniform RV on $(0, 1),$ $Q(u)$ is the quantile function of the RV $Z,$ and $q(u)$ is the quantile density function, i.e., $q(u) = \frac{d}{du}Q(u).$

Proof. Using (1.4) and (1.7), then WCRTE is provided by

Let $Q(u) = F^{-1}_Z(z)$ be the quantile function. By using the well-known relation $q(u)f_Z(Q(u)) = 1$ in (2.1), we get

□

Remark 2.1. If $m = 0$ and $k = 1,$ the WCRTE of the concomitant of the $r$th OS, $Z_{[r, n, 0, 1]}: = Z_{[r:n]},$ based on the FGM family is given by

where $\phi_{(r:n)} = \frac{n-2r+1}{n+1}$ (cf. ^[30]).

Remark 2.2. The model of record values is a special case of the $m$-GOSs by putting $m = -1$ and $k = 1.$ Therefore, the WCRTE of the concomitant $Z_{[n]}$ of the $n$th upper record value based on the FGM family is given by

where $\delta_{(n)} = 2^{-(n-1)}-1$.

Example 2.1. Consider the two variables, $Y$ and $Z,$ that possess exponential distribution (ED) from the FGM family (represented by FGM-ED) (i.e., $F_{Z}(z) = 1-e^{-\lambda z}, z, \lambda > 0$). Then

Based on (2.2), we get the WCRTE in $Z_{[r, n, m, k]}$ as follows:

Example 2.2. Consider $Y$ and $Z$ to be Weibull distributions (WD) derived from the FGM family (i.e., $F_{Z}(z) = 1-e^{-(\frac{z}{\beta})^{\eta}}, z > 0, \; \eta, \beta > 0$). Then

Thus, $\zeta^{w}_{\theta, \alpha}(Z_{[r, n, m, k]})$ is given by

Tables 1 and 2 show aspects of the behavior of the WCRTE for $Z_{[r:n]}$ and $Z_{[n]},$ respectively. In Tables 1 and 2, the following properties can be extracted:

● Generally, $\zeta^{w}_{\theta, \alpha}(Z_{[r:n]}) = \zeta^{w}_{\theta, -\alpha}(Z_{[n-r+1:n]}).$ Also, the value of $\zeta^{w}_{\theta, \alpha}(Z_{[r:n]})$ slowly increases as the value of $n$ increases (cf. Table 1).

● We see that the value of $\zeta^{w}_{\theta, -\alpha}(Z_{[n]})$ goes up as $n$ goes up, the value of $\zeta^{w}_{\theta, \alpha}(Z_{[n]})$ goes down as $n$ goes up, and it almost stays the same when $n = 15$ (cf. Table 2).

Remark 2.3. It is worth noting that the relation $\zeta^{w}_{\theta, \alpha}(Z_{[r:n]}) = \zeta^{w}_{\theta, -\alpha}(Z_{[n-r+1:n]})$ may be theoretically proved for every $\theta > 0,$ $-1\leq\alpha\leq1,$ and any non-negative continuous RV $Z$ by applying Theorem 2.1 and remarking that $C^{*}_{(n-r+1, n, 0, 1)} = - C^{*}_{(r, n, 0, 1)}.$

2.1.   AWCRTE for the concomitant of the $m-$GOS of order $\theta$

Theorem 2.2. The AWCRTE for the concomitant $Z_{[r, n, m, k]}$ of the $m-$GOS based on the FGM family is given by

Proof. Using (1.4) and (1.8), we get

□

Remark 2.4. If $m = 0$ and $k = 1,$ the AWCRTE of the concomitant $Z_{[r:n]}$ of the $r$th OS based on the FGM family is given by

Remark 2.5. If $m = -1$ and $k = 1,$ the AWCRTE of concomitant of the $n$th upper record value based on the FGM family is given by

In survival analysis, if an RV $T^*$ given $\Theta = \theta$ has a DF $F(t|\theta),$ the weighted mean residual lifetime (WMRL) of $T^*$ given $\Theta = \theta$ is then obtained as

which is valid for all $\theta$ that $\overline F(t|\theta) > 0$ and also plays a crucial role in survival analysis. Now, we provide a relationship between the WMRL, $m^{*}_{F_{[r, n, m, k]}}(t) = \int_{t}^{\infty}z\frac{ \overline F_{[r, n, m, k]}(z)}{ \overline F_{[r, n, m, k]}(t)}dz, \; \overline F_{[r, n, m, k]}(t) > 0,$ and the AWCRTE of an RV $Z.$ This relationship illustrates AWCRTE's crucial role in survival analysis.

Lemma 2.1. For a non-negative continuous RV $Z$ with SF $\overline F_{Z}(z),$

Proof. First, we note $\frac{d}{dz}(m^{*}_{F_{[r, n, m, k]}}(z) \overline F_{[r, n, m, k]}(z)) = -z \overline F_{[r, n, m, k]}(z).$ Thus, by using the fact in (1.8), we get

Now the result follows by using integration by parts. □

Example 2.3. Consider the RVs, $Y$ and $Z,$ that follow FGM-ED. Based on (2.3), we get the AWCRTE in $Z_{[r, n, m, k]}$ as follows:

Moreover, it is easy to check that

which yields

Example 2.4. Consider $Y$ and $Z$ to be WD derived from the FGM family. Then $\int_{0}^{\infty}z(e^{-(\frac{z}{\beta})^{\eta}})dz = \frac{\Gamma(\frac{2}{\eta})\beta^{2}}{\eta},$

and

Therefore, the measure $\xi^{w}_{\theta, \alpha}(Z_{[r, n, m, k]})$ is given by

Example 2.5. Suppose $Y$ and $Z$ are continuous RVs with respective PDF $f_{Y}(y) = \frac{1}{a}, \; 0 < y < a,$ and $g_{Z}(z) = \frac{1}{a}, \; h < z < a+h, \; h > 0.$ From (1.5), we have

Now, from (1.8), we get

and

Therefore, although ${\cal{E}}_{\theta, \alpha}(Y_{[r, n, m, k]}) = {\cal{E}}_{\theta, \alpha}(Z_{[r, n, m, k]}),$ $\xi^{w}_{\theta, \alpha}(Y_{[r, n, m, k]})\neq\xi^{w}_{\theta, \alpha}(Z_{[r, n, m, k]}).$

Example 2.6. Suppose $Z$ has a Pareto distribution with DF $F_{Z}(z) = 1-(\frac{b}{z})^{a}, z\geq b, b > 0, a > 0.$ Then

Also, we can show that

and

The following lemma shows that $\xi^{w}_{\theta, \alpha}(Z_{[r, n, m, k]})$ is a shift-dependent measure.

Lemma 2.2. Let $Z = aX+b$ with $a > 0$ and $b\geq0.$ Then

Proof. First, since the linear transformation, with $b\geq 0,$ preserves the order relation, then for every $1\leq r\leq n,$ we get $Z_{r, n, m, k} = aX_{r, n, m, k}+b$ and $Z_{[r, n, m, k]} = aX_{[r, n, m, k]}+b.$ The proof follows using the fact that $\overline F_{Z_{[r, n, m, k]}}(u) = \overline F_{X_{[r, n, m, k]}}(\frac{u-b}{a}).$ □

As shown in Tables 3–6 of the FGM-ED and FGM-WD, respectively, the AWCRTE for $Z_{[r:n]}$ and $Z_{[n]}$ are presented. After running the numbers through MATHEMATICA version 12, we can deduce the following properties from Tables 3–6.

● Generally, $\xi^{w}_{\theta, \alpha}(Z_{[r:n]}) = \xi^{w}_{\theta, -\alpha}(Z_{[n-r+1:n]})$ (similar to Remark 2.3, this symmetry relation can easily be proved theoretically). Also, the value of $\xi^{w}_{\theta, \alpha}(Z_{[r:n]})$ slowly increases as the value of $n$ increases (see Tables 3 and 5).

● We can see that the value of $\xi^{w}_{\theta, -\alpha}(Z_{[n]})$ increases as $n$ increases, the value of $\xi^{w}_{\theta, \alpha}(Z_{[n]})$ decreases as $n$ decreases, and it almost stays the same when $n = 15$ (see Table 4).

● At most, the value of $\xi^{w}_{\theta, -\alpha}(Z_{[n]})$ increases as $n$ increases, but the value of $\xi^{w}_{\theta, \alpha}(Z_{[n]})$ decreases as $n$ increases, and it almost stays the same when $n = 17$ (see Table 6).

2.2.   Dynamic AWCRTE of order $\theta$

When doing a reliability analysis, one crucial quantity to consider is the component or system's residual lifetime. Assuming a component $Z$ has survived for a certain amount of time $t,$ its residual lifetime is $Z_t = (Z-t)|Z > t.$ This is the AWCRTE with residual lifetime $Z_t,$ which is dynamic and has order $\theta.$ Then, the dynamic AWCRTE in the $r$th concomitant based on the FGM family is given by

In the following theorem, we establish the connection between the dynamic AWCRTE and WMRL.

Theorem 2.3. Let $Z$ be an absolutely continuous non-negative RV with WMRL function $m^{*}_{F_{[r, n, m, k]}}(t),$ and then,

Proof. From (2.4), we have

The proof is complete. □

3.   Weighted cumulative past Tsallis entropy of order $\theta$

Here, we study the WCPTE and its dynamic version in concomitant $Z_{[r, n, m, k]},$ from the FGM family, with numerical illustrations according to the sub-model OSs and record values.

Theorem 3.1. The WCPTE for the $r$th concomitant of $m-$GOSs based on the FGM family is given by

Proof. Using (1.3) and (1.6), then the WCPTE is provided by

□

Remark 3.1. Let $F_{[r:n]}(z)$ be the DF of the $r$th concomitant $Z_{[r:n]}.$ By putting $m = 0$ and $k = 1,$ then the WCPTE in the $r$th concomitant of the OSs is given by

Remark 3.2. Let $F_{[n]}(z)$ be the DF of the $n$th upper record. By putting $k = 1$ and $m = -1,$ then the WCPTE in the concomitant of the $n$th upper record value is given by

The link between the weighted mean past lifetime (WMPL), $\mu^{*}_{F_{[r, n, m, k]}}(t) = \int_{0}^{t}z\frac{ F_{[r, n, m, k]}(z)}{ F_{[r, n, m, k]}(t)}dz,$ $F_{[r, n, m, k]}(t) > 0,$ and the WCPTE is then determined.

Lemma 3.1. Let $Z$ be a non-negative continuous RV with DF $F_Z(z),$ and then

Proof. We have $\frac{d}{dz}(\mu^{*}_{F[r, n, m, k]}(z) F_{[r, n, m, k]}(z)) = z F_{[r, n, m, k]}(z).$ Using (3.1), we get

□

Example 3.1. Assume that the uniform distribution (UD) of $Y$ and $Z$ results from the FGM family (i.e., $F_{Z}(z) = z, 0\leq z\leq1$). After simple algebra, we get

and

Then, based on (3.1), this leads to the following WCPTE in $Z_{[r, n, m, k]}$:

and

We can easily show that

Example 3.2. Let $Y$ and $Z$ be two variables that represent power distributions obtained from the FGM family (i.e., $F_{Z}(z) = z^{c}, \; 0\leq z\leq1, c > 0$). Then

and

Thus, based on (3.1), this leads to the following WCPTE in $Z_{[r, n, m, k]}$:

Lemma 3.2. If $Z = aX+b$ with $a > 0$ and $b\geq0,$ then

Proof. The proof follows using the fact that $F_{Z_{[r, n, m, k]}}(u) = F_{X_{[r, n, m, k]}}(\frac{u-b}{a}).$ □

Tables 7 and 8 show some aspects of the behavior of the WCPTE for $Z_{[n]}$ and $Z_{[r:n]}$ based on the FGM-UD. From Tables 7 and 8, the ensuing characteristics are extractable:

● We see that the value of ${\cal{P}}^{w}_{\theta, \alpha}(Z_{[n]})$ increases as $n$ increases, the value of ${\cal{P}}^{w}_{\theta, -\alpha}(Z_{[n]})$ decreases as $n$ increases, and it almost stays the same when $n = 15$ (see Table 7).

● Generally, ${\cal{P}}^{w}_{\theta, \alpha}(Z_{[r:n]}) = {\cal{P}}^{w}_{\theta, -\alpha}(Z_{[n-r+1:n]}).$ Also, the value of ${\cal{P}}^{w}_{\theta, \alpha}(Z_{[r:n]})$ slowly increases at $r\leq (\frac{n}{2}+1),$ and decreases at $r\geq (\frac{n}{2}+1)$ (see Table 8).

Dynamic WCPTE measure

Dynamic WCPTE (DWCPTE) of an RV $Z$ is the WCPTE of the RV $[t-Z|Z < t], t > 0.$ The DWCPTE variant from the FGM family in $Z_{[r, n, m, k]}$ is provided as

Theorem 3.2. Let $Z$ be an absolutely continuous non-negative RV with WMPL function $\mu^{*}_{F[r, n, m, k]}(t),$ and then,

Proof. The Proof is similar to (2.5). □

4.   Empirical AWCRTE and WCPTE

The issue of estimating the AWCRTE and WCPTE for concomitant $Z_{[r, n, m, k]}$ utilizing the empirical AWCRTE will be examined next. For every $i = 1, 2, ..., n,$ consider the FGM sequence $(Y_i, Z_i).$ In accordance with (2.3), the empirical AWCRTE of the set $Z _{[r, n, m, k]}$ can be computed as follows:

where for any DF $F(.),$ the symbol $\hat F(.)$ stands for the empirical DF of $F(.)$ and $\Delta_{j} = \frac{Z^{2}_{(j+1)}-Z^{2}_{(j)}}{2},$ $j = 1, 2..., n-1,$ are the sample spacings based on ordered random samples of $Z_{j}.$ Similarly, based on (3.1), the empirical WCPTE of the set $Z _{[r, n, m, k]}$ can be expressed as

Example 4.1. Suppose that $Z$ has a Rayleigh distribution with PDF $f_Z(z) = 2\lambda ze^{-\lambda z^{2}}, z > 0, \lambda > 0.$ Then, $Z^{2}$ has ED with mean $\frac{1}{\lambda}$ and $\Delta_{j} = \frac{Z^{2}_{(j+1)}-Z^{2}_{(j)}}{2}$ has ED with mean $\frac{1}{2\lambda(n-j)}, j = 1, 2..., n-1.$ The expected value and variance of the empirical AWCRTE in $\hat{\xi}^{w}_{\theta, \alpha}(Z_{[r, n, m, k]})$ are given by

and

Example 4.2. Let $(Y_i, Z_i), i = 1, 2, ..., n,$ be a random sample from the FGM family with PDF $f_Z(z) = 2z, \; 0 < z < 1.$ Then, $Z^{2}$ has a standard uniform distribution. Furthermore, $\Delta_{j} = \frac{Z^{2}_{(j+1)}-Z^{2}_{(j)}}{2}$ follows the beta distribution with mean $\frac{1}{2(n+1)}$ and variance $\frac{n}{4(n+1)^{2}(n+2)}.$ The mean and variance of the empirical WCPTE in $\hat{{\cal{P}}}^{w}_{\theta, \alpha}(Z_{[r, n, m, k]})$ are given by

and

Both the AWCRTE and empirical AWCRTE in $Z_{[r:n]}$ from FGM-ED at $n = 50,$ as well as the WCPTE and empirical WCPTE in $Z_{[r:n]}$ from FGM-UD, are shown in Figures 2 and 3. Figures 2 and 3 can be utilized to ascertain the subsequent properties:

$1)$ Mostly, the AWCRTE and the empirical AWCRTE are close together.

$2)$ Generally, the WCPTE and the empirical WCPTE are very close.

5.   Real data application

As we have previously highlighted, Tsallis entropy is a generalization of Shannon entropy that works better with non-extensive systems. This makes it possible to analyze complex data, like those found in medical settings, such as electroencephalogram (EEG) signals used to diagnose epilepsy, in a more flexible and reliable manner. Tsallis entropy offers a more nuanced view of the system's non-linear and non-extensive dynamics, providing insights that may not be captured by Shannon entropy. EEG signals are an example of a complicated medical data set used in epilepsy diagnosis.

Using Tsallis entropy and a few of its associated measures that are covered in this article, we examine two medical data sets below. Tsallis entropy provides information that Shannon entropy might miss, despite the fact that it is challenging to practically verify the level of complexity and non-linearity of the handled systems.

Example 5.1. We use the data for 30 patients from ^[31]. Let $Y$ refer to the first recurrence time and $Z$ to the second recurrence time, as follows: $\text{Y}$ is (8, 23, 22,447, 30, 24, 7,511, 53, 15, 7,141, 96,149,536, 17,185,292, 22, 15,152,402, 13, 39, 12,113,132, 34, 2,130) and $Z$ is (16, 13, 28,318, 12,245, 9, 30,196,154,333, 8, 38, 70, 25, 4,117,114,159,108,362, 24, 66, 46, 40,201,156, 30, 25, 26). Reference ^[32] introduced FGM bivariate WD, and they discussed the estimation of the parameters of this model and found the maximum likelihood estimates (MLEs) of the shape and scale parameters $(\eta_i, \beta_i), i = 1, 2,$ as $(0.75106,100.11993)$ and $(0.92435, 98.24665),$ respectively, and $\alpha = 0.34801.$ To find a trust region or confidence intervals for the parameters of the FGM bivariate WD, we can further use, among many other methods, the fisher information matrix to derive asymptotic confidence intervals (which provides the standard method). However, the primary objective is to determine whether the FGM-WD model, after estimating its unknown parameters, adequately fits the given data and then examines the AWCRTE and WCRTE.

Table 9 examines the AWCRTE and WCRTE for FGM-WD (0.75106,100.11993, 0.92435, 98.24665). For the concomitants $Z_{[r:30]}, r = 1, 2, 14, 15, 29, 30,$ i.e., the lower and upper extremes concomitants, and the central values concomitants. We observe that the $\xi^{w}_{\theta, \alpha}(Z_{[r:30]})$ and $\zeta^{w}_{\theta, \alpha}(Z_{[r:30]})$ have maximum values at extremes. Figures 4 and 5 provide a fundamental statistical analysis illustrating the data.

Example 5.2 (Cholesterol data set). This data set includes cholesterol levels measured at 5 and 25 weeks after treatment in 30 patients (see ^[33]). We fit the data based on FGM-WD $(\eta_1, \beta_1;$ $\eta_2, \beta_2).$ The MLEs of parameters are $\widehat{\eta}_1 = 2.93893, \widehat{\beta}_1 = 1.21085, \widehat{\eta}_2 = 2.589,$ $\widehat{\beta}_2 = 1.10099,$ and $\widehat{\alpha} = 1$. Table 10 examines the AWCRTE and WCRTE for FGM-WD (2.93893, 1.21085, 2.589, 1.10099). For the concomitants $Z_{[r:30]}, r = 1, 2, 14, 15, 29, 30,$ i.e., the lower and upper extremes concomitants, and the central values concomitants. We observe that the $\xi^{w}_{\theta[r:30]}(Z)$ and $\zeta^{w}_{\theta[r:30]}(Z)$ have maximum values at extremes.

6.   Conclusions

Among bivariate distributions, the FGM model is one of the most well-known and useful in recent years. The FGM bivariate distribution family has been widely accepted for many practical applications. Furthermore, the recently introduced redundant concomitants of OSs have regained popularity due to their usefulness in prediction and selection contexts. Another important concept that has gained attention is Tsallis entropy, which has been applied in various fields, including physics and chemistry. Every year, new applications of these measures are discovered. Some of the most significant related measures recently introduced are WCRTE and WCPTE. AWCRTE, WCPTE, and their dynamic counterparts for the concomitants of $m$-GOSs were derived from the FGM bivariate family. The characteristics of the proposed entropy measures were analyzed. These entropy measures were used to characterize the exponential and Pareto distributions. Applications of these findings were presented for OS and record values with uniform, Weibull, and exponential marginal distributions. Additionally, non-parametric estimators of AWCRTE and WCPTE were proposed for calculating the new information measures.

Two real-world data sets were evaluated for illustration, yielding satisfactory results. The parameter $\theta$ in Tsallis entropy controls the sensitivity of the entropy to rare events and affects the system's "distributional" properties. Specifically, when $\theta = 1,$ it corresponds to the standard Shannon entropy, while $\theta > 1$ and $\theta < 1$ modify the sensitivity to the tail and central parts of the distribution, respectively.

To estimate $\theta$ empirically, the Tsallis model is typically fitted to data using methods such as maximum likelihood estimation, least squares fitting, or numerical optimization. For more details, see ^[34]. However, in this study, we did not address the estimation of $\theta$. Instead, we selected different values of $\theta$ to demonstrate how the results change as the value of $\theta$ varies.

Validating the quality of estimates for unknown parameters in real data cases can be quite challenging and often requires specialized techniques like bootstrapping. However, this aspect is beyond the scope of our current study. It is important to note that parameter estimation is just one part of our statistical analysis. The primary objective is to determine whether the FGM-WD model, after estimating its unknown parameters, adequately fits the given data, which was confirmed in the examples we explored. In future work, where we will conduct an in-depth study of a specific practical situation, we plan to address this issue more thoroughly.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors are grateful to the editor and anonymous referees for their insightful comments and suggestions, which helped to improve the paper's presentation. This research was conducted under a project titled "Ongoing Research Funding Program", funded by King Saudi University, Riyadh, Saudi Arabia, under grant number (ORF-2025-969).

Conflict of interest

The authors declare no conflict of interest.