*4.1. Shannon and Shannon-Related Entropies*

One can easily notice that the relationship between Shannon–Awad entropy (18) and Shannon entropy (12) is:

$$H^{SA}(X) = H^S(X) + \log \delta. \tag{34}$$

In the following, we provide natural extensions of the results obtained in [24], considering Shannon–Awad entropy instead of Shannon entropy. Thus, **Shannon–Awad entropy** of the concomitant of the r-th GOS from the FGM family is:

$$H^{SA}(Y\_{[r]}^\*) = \mathbb{W}(r, a, n, m, k) + (H^{SA}(Y) - \log \delta)(1 - a\mathbb{C}\_r^\*) - 2a\mathbb{C}\_r^\*\phi\_f(\mu) + \log \delta^{[r]},\tag{35}$$

where *W*(*r*, *α*, *n*, *m*, *k*) and *φ<sup>f</sup>* are given by (14) and (15) and

$$\delta = \sup \{ f\_Y(\mathbf{x}) | \mathbf{x} > 0 \}, \quad \delta^{[r]} = \sup \{ \mathcal{g}\_{[r]}(\mathbf{x}) | \mathbf{x} > 0 \}.$$

**Remark 5.** *For the simple OS, the r-th OS concomitant from the FGM family, Shannon–Awad entropy is:*

$$\begin{split} H^{SA}(\boldsymbol{Y}\_{[r]}) &= W(\boldsymbol{r}, \boldsymbol{\alpha}, \boldsymbol{n}, \boldsymbol{0}, 1) + (H^{SA}(\boldsymbol{Y}) - \log \delta) \left( 1 + \boldsymbol{n} \frac{\boldsymbol{n} - 2\boldsymbol{r} + 1}{n + 1} \right) + \\ &+ 2\boldsymbol{n} \frac{\boldsymbol{n} - 2\boldsymbol{r} + 1}{n + 1} \phi\_f(\boldsymbol{u}) + \log \delta\_{[r]}, \end{split} \tag{36}$$

*with δ*[*r*] = sup{*g*[*r*](*x*)|*x* > 0} *and g*[*r*] *being here the pdf of the concomitant r-th-order statistics.*

*For the record values, the r-th record value concomitant from the FGM family, Shannon–Awad entropy is:*

$$\begin{split} H^{SA}(R\_{[r]}) &= \mathcal{W}(r, a, n\_r - 1, 1) + (H^{SA}(Y) - \log \delta) \Big[ 1 + a(2^{1-r} - 1) \Big] + \\ &+ 2a(2^{1-r} - 1)\phi\_f(u) + \log \delta\_{[r]} \end{split} \tag{37}$$

*with δ*[*r*] = sup{*g*[*r*](*x*)|*x* > 0} *and g*[*r*] *being here the pdf of the concomitant r-th record value.*

*In the case of progressive type II censoring order statistics with equi-balanced censoring scheme, the Shannon–Awad entropy of the concomitant of r-th-order statistic from the FGM family is* (35) *with m* = *R, the removal number.*

An extension of the above entropies are **residual and past Shannon–Awad entropies**. We define residual Shannon–Awad entropy as:

$$H^{SA}(X;t) = -E\left[\log\frac{f(X)}{\overline{F}(t)\delta^F\_{(t,\infty)}} \middle| X > t\right],\tag{38}$$

where

*δF*¯ (*t*,∞) <sup>=</sup> <sup>1</sup> *F*(*t*) sup{ *f*(*x*)|*x* ∈ (*t*, ∞)}. (39)

In terms of failure rate, residual Shannon–Awad entropy can be obtained:

$$H^{SA}(X;t) = 1 - E[\log \lambda(X) | X > t] + \log \delta^{\mathbf{f}}\_{(t,\infty)}.\tag{40}$$

We notice that the relationship between residual Shannon entropy and residual Shannon–Awad entropy is similar to (34) and it is:

$$H^{SA}(X;t) = H^S(X;t) + \log \delta^{\rm f}\_{(t,\infty)}.\tag{41}$$

In a similar way, we can extend past Shannon entropy to past Shannon–Awad entropy:

$$\overline{H}^{SA}(X;t) = -E\left[\log \frac{f(X)}{F(t)\delta\_{(0,t)}^F} | X < t\right],\tag{42}$$

where

$$\delta\_{(0,t)}^F = \frac{1}{F(t)} \sup \{ f(\mathbf{x}) | \mathbf{x} \in (0, t) \}. \tag{43}$$

As a function of reversed failure rate, past Shannon–Awad entropy can be written:

$$H^{SA}(X;t) = 1 - E[\log \tau(X) | X < t] + \log \delta^F\_{(0,t)}.\tag{44}$$

We can write also the relationship between past Shannon–Awad entropy and past Shannon entropy:

$$
\overline{H}^{SA}(X;t) = \overline{H}^S(X;t) + \log \delta\_{(0,t)}^F. \tag{45}
$$

Taking into account the above relationships, (21), and (26), we can obtain the Awadtype extension of the Shannon entropy for the concomitant of r-th GOS from the FGM family, when the concomitant represents the residual life or the past life of a unit.

**Theorem 1.** *Residual Shannon–Awad entropy for the concomitant of r-th GOS from the FGM family is:*

$$\mathcal{H}^{SA}(Y\_{[r]}^{\vartri};t) = \log \overline{\mathcal{G}}\_{[r]}(t) - \frac{1}{\overline{\mathcal{G}}\_{[r]}(t)} \left\{ (1 - a\mathcal{C}\_r^{\vartri}) [\overline{F}\_Y(t)(\log \overline{F}\_Y(t) - H^{SA}(Y; t) + \delta\_{l,m}^{\overline{\mathcal{G}}\_Y})] + \\ \tag{46}$$

$$+ 2a\mathcal{C}\_r^{\vartri}\phi\_f(t) + K\_1(r, t, a, n, m, k) \right\} \\ + \log \overline{\mathcal{S}}\_{(t, \mathbb{R}^d)}^{\mathcal{G}\_{[r]}}$$

*where K*1(*r*, *t*, *α*, *n*, *m*, *k*) *and φf*(*t*) *are given by* (22)*,* (23) *respectively, and*

$$\delta\_{(t,\infty)}^{\mathbb{F}\_Y^\*} = \frac{1}{\overline{F}\_Y(t)} \sup \{ f\_Y(\mathbf{x}) | \mathbf{x} \in (t,\infty) \}, \quad \delta\_{(t,\infty)}^{\mathbb{G}\_{[r]}} = \frac{1}{\overline{G}\_{[r]}(t)} \sup \{ g\_{[r]}(\mathbf{x}) | \mathbf{x} \in (t,\infty) \}.$$

*Past Shannon–Awad entropy for the concomitant of r-th GOS from the FGM family is:*

$$\begin{split} \overline{H}^{SA}(Y\_{[r]}^{\*};t) &= \log \mathcal{G}\_{[r]}(t) - \frac{1}{\widehat{\mathcal{G}}\_{[r]}(t)} \left\{ (1 - a \mathcal{C}\_{r}^{\*}) [F\_{Y}(t)(\log \mathcal{F}\_{Y}(t) - \overline{\mathcal{H}}^{\rm SA}(Y;t) + \boldsymbol{\delta}^{\rm F\_{Y}}\_{(0,t)})] + \\ &\quad + 2a \mathcal{C}\_{r}^{\*} \overline{\widehat{\mathcal{G}}}\_{f}(t) + K\_{\mathbb{S}}(r,t,a,n,m,k) \right\} + \log \boldsymbol{\delta}^{\rm G\_{[r]}}\_{(0,t)} \end{split} \tag{47}$$

*where K*2(*r*, *t*, *α*, *n*, *m*, *k*) *and φf*(*y*)*, and δ*(0,*t*) *are given by* (22) *and* (23) *respectively, and*

$$\delta\_{(0,t)}^{F\_Y} = \frac{1}{F\_Y(t)} \sup \{ f\_Y(\mathbf{x}) | \mathbf{x} \in (0,t) \}, \quad \delta\_{(0,t)}^{G\_{[r]}} = \frac{1}{G\_{[r]}(t)} \sup \{ g\_{[r]}(\mathbf{x}) | \mathbf{x} \in (0,t) \}.$$

**Corollary 1.** *The residual Shannon–Awad entropy for the concomitant of r-th-order statistic is:*

$$H^{SA}(Y\_{[r]};t) = \log \mathsf{G}\_{[r]}(t) - \frac{1}{\mathsf{G}\_{[r]}(t)} \left\{ \left(1 + a\frac{n-2r+1}{n+1}\right) [\mathsf{F}\_Y(t)(\log \mathsf{F}\_Y(t) - H^{SA}(Y;t) + \delta\_{l,\infty}^{\tilde{\Gamma}\_{[r]}})] + \dots \right\}$$

$$-2a\frac{n-2r+1}{n+1}\phi\_f(t) + K\_1(r,t,a,n,0,1)\Big{)} + \log \delta\_{(t,\infty)}^{\tilde{\mathcal{C}}\_{[r]}}.$$

*The residual Shannon–Awad entropy for the concomitant of r-th record value is:*

$$H^{SA}(R\_{[r]};t) = \log \overline{\mathbf{C}}\_{[r]}(t) - \frac{1}{\overline{\mathbf{C}}\_{[r]}(t)} \left\{ (1 + a(2^{1-r}-1)) [\overline{\mathbf{F}}\_{Y}(t)(\log \overline{\mathbf{F}}\_{Y}(t) - H^{SA}(Y;t) + \delta^{\mathbf{F}\_{Y}}\_{t,\infty})] + \right. \tag{49}$$

$$-2a(2^{1-r}-1)\phi\_{f}(t) + \mathbf{K}\_{1}(r,t,u,n\_{r}-1,1) \Big\} + \log \delta^{\mathbf{C}\_{[r]}}\_{(t,\infty)}.$$

*In the case of progressive type II censoring order statistics with equi-balanced censoring scheme, the residual Shannon–Awad entropy of the concomitant of r-th-order statistic from the FGM family is* (46) *with m* = *R, the removal number.*

*Similar results can be obtained also for past Shannon–Awad entropy.*
