*4.2. Tsallis and Tsallis-Related Entropies*

Information measures related to Tsallis entropy for the concomitants are very few in the literature. In [35], Tsallis entropy and residual Tsallis entropy for the concomitants of the record values from the FGM family are obtained. In this subsection, we will obtain more general results, computing Tsallis entropies for the concomitants of generalized order statistics and, furthermore, considering Awad-type extensions of the Tsallis entropies.

**Theorem 2.** *Tsallis entropy for the concomitant of the r-th GOS from the FGM family is:*

$$H^{T}(Y\_{[r]}^{\*}) = \frac{1}{q-1} \left\{ 1 - \sum\_{k=0}^{q} \sum\_{s=0}^{k} (-1)^{s} \binom{q}{k} \binom{k}{s} 2^{k-s} a^{k} \mathcal{C}\_{r}^{\*k} E\_{U} \left[ f\_{Y} (F\_{Y}^{-1}(\mathcal{U}))^{q-1} \mathcal{U}^{k-s} \right] \right\}, \tag{50}$$

*where U is an U*(0, 1) *random variable and EU is the expectation of fY*(*F*−<sup>1</sup> *<sup>Y</sup>* (*U*))*q*−1*Uk*−*<sup>s</sup> .* **Proof.** Taking into account the definitions of Tsallis entropy (29) and the density of the concomitants (6), we obtain:

$$\begin{aligned} H^T(Y\_{[r]}^\*) &= \frac{1}{q-1} \left\{ 1 - \int\_0^\infty [g\_{[r]}(y)]^q dy \right\} = \\ &= \frac{1}{q-1} \left( 1 - \int\_0^\infty [f\_Y(y)]^q (1 + aC\_r^\*(2F\_Y(y) - 1))^q dy \right). \end{aligned}$$

We have that:

$$(1+a\mathbb{C}\_r^\*(2F\_Y(y)-1))^q = \sum\_{k=0}^q \sum\_{s=0}^k (-1)^s \binom{q}{k} \binom{k}{s} 2^{k-s} a^k \mathbb{C}\_r^{\*k} [F\_Y(y)]^{k-s}.$$

Additionally, if we consider the transformation

$$F\_Y(y) = u; \ y = F\_Y^{-1}(u), \ f\_Y(y) dy = du\_\prime y$$

the result (50) follows.

**Corollary 2.** *The Tsallis entropy for the concomitant of r-th-order statistic is:*

$$H^{T}(Y\_{[r]}) = \frac{1}{q-1} \left\{ 1 - \sum\_{k=0}^{q} \sum\_{s=0}^{k} (-1)^{s} \binom{q}{k} \binom{k}{s} 2^{k-s} a^{k} (-1)^{k} \left( \frac{n-2r+1}{n+1} \right)^{k} \times \\ \qquad \text{(51)}$$

$$\times E\_{II} \left[ f\_{Y}(F\_{Y}^{-1}(\mathcal{U}))^{q-1} \mathcal{U}^{k-s} \right] \right\}.$$

*The Tsallis entropy for the concomitant of r-th record value is:*

$$H^{T}(R\_{\left[r\right]}) = \frac{1}{q-1} \left\{ 1 - \sum\_{k=0}^{q} \sum\_{s=0}^{k} (-1)^{s} \binom{q}{k} \binom{k}{s} 2^{k-s} a^{k} (-1)^{k} \left(2^{1-r} - 1\right)^{k} \times \\ \tag{52}$$

$$\times E\_{II} \left[ f\_{Y} (F\_{Y}^{-1}(\mathcal{U}))^{q-1} \mathcal{U}^{k-s} \right] \right\}.$$

*In the case of progressive type II censoring order statistics with equi-balanced censoring scheme, the Tsallis entropy of the concomitant of r-th-order statistic from the FGM family is* (50) *with m* = *R, the removal number.*

We now discuss some Tsallis-related entropies. First, we give an Awad-type extension of Tsallis entropy and then we focus on Residual Tsallis, Past Tsallis entropies and their Awad-type extensions.

Several Awad-type extensions have been proposed in the literature ([51,52]). Now, we introduce this type of extension for Tsallis entropy and we define **Tsallis–Awad entropy** for a continuous random variable *X* which take values in R:

$$H^{TA}(X) = \frac{1}{q-1} \left\{ 1 - \int\_{-\infty}^{+\infty} \left[ \frac{f(\mathbf{x})}{\delta} \right]^{q-1} f(\mathbf{x}) d\mathbf{x} \right\} = \frac{1}{q-1} \left\{ 1 - \frac{1}{\delta^{q-1}} \int\_{-\infty}^{+\infty} [f(\mathbf{x})]^q d\mathbf{x} \right\},\tag{53}$$

where *<sup>δ</sup>* <sup>=</sup> sup{ *<sup>f</sup>*(*x*)|*<sup>x</sup>* <sup>∈</sup> <sup>R</sup>}.

We notice that the relationship between Tsallis–Awad entropy and Tsallis entropy is:

$$H^{TA}(X) = \delta^{1-q} H^T(X) + \log\_q \delta. \tag{54}$$

Using (50) and (54), we can obtain the expression of Tsallis–Awad entropy for the concomitant of the r-th GOS from the FGM family:

**Theorem 3.** *Tsallis–Awad entropy for the concomitant of the r-th GOS from the FGM family is:*

$$H^{TA}(Y\_{[r]}^{\*}) = \frac{\delta^{1-q}}{q-1} \left\{ 1 - \sum\_{k=0}^{q} \sum\_{s=0}^{k} (-1)^{s} \binom{q}{k} \binom{k}{s} 2^{k-s} \alpha^{k} \mathcal{C}\_{r}^{\*k} E\_{\mathcal{U}} \left[ f\_{Y} (F\_{Y}^{-1}(\mathcal{U}))^{q-1} \mathcal{U}^{k-s} \right] \right\} + \log\_{q} \delta,\tag{55}$$

*where U is an U*(0, 1) *random variable, and, in this case, δ* = sup{*g*[*r*](*x*)|*x* > 0}*.*

**Corollary 3.** *The Tsallis–Awad entropy for the concomitant of r-th-order statistic is:*

$$H^T(Y\_{[r]}) = \frac{\delta^{1-q}}{q-1} \left\{ 1 - \sum\_{k=0}^q \sum\_{s=0}^k (-1)^s \binom{q}{k} \binom{k}{s} 2^{k-s} a^k (-1)^k \left( \frac{n-2r+1}{n+1} \right)^k \times \right. \tag{56}$$

$$\times E\_{\mathcal{U}} \left[ f\_Y(F\_Y^{-1}(\mathcal{U}))^{q-1} \mathcal{U}^{k-s} \right] \Big) + \log\_q \delta.$$

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

$$H^{T}(R\_{\left[r\right]}) = \frac{\delta^{1-q}}{q-1} \left\{ 1 - \sum\_{k=0}^{q} \sum\_{s=0}^{k} (-1)^{s} \binom{q}{k} \binom{k}{s} 2^{k-s} a^{k} (-1)^{k} \left(2^{1-r} - 1\right)^{k} \times \\ \tag{57}$$

$$\times E\_{lI} \left[ f\_{Y}(F\_{Y}^{-1}(lI))^{q-1} U^{k-s} \right] \right\} + \log\_{q} \delta.$$

*In the case of progressve type II censoring order statistics with equi-balanced censoring scheme, the Tsallis–Awad entropy of the concomitant of r-th-order statistic from the FGM family is* (55) *with m* = *R, the removal number.*

In a similar way to the definition of **residual Tsallis** (31), we can consider the **past Tsallis entropy**:

$$\overline{H}^T(\mathbf{X};t) = \frac{1}{q-1} \left\{ 1 - \int\_0^t \left[ \frac{f(\mathbf{x})}{F(t)} \right]^q d\mathbf{x} \right\} = \frac{1}{q-1} \left\{ 1 - \frac{1}{F(t)^q} \int\_0^t [f(\mathbf{x})]^q d\mathbf{x} \right\}.\tag{58}$$

Taking into account Theorem 2, the following theorem is naturally deduced:

**Theorem 4.** *Residual Tsallis entropy for the concomitant of the r-th GOS from the FGM family is:*

$$H^{T}(Y\_{[r]}^{\*};t) = \frac{1}{q-1} \left\{ 1 - \frac{1}{\left(\bar{\mathcal{G}}\_{[r]}(t)\right)^{q}} \sum\_{k=0}^{q} \sum\_{s=0}^{k} (-1)^{s} \binom{q}{k} \binom{k}{s} 2^{k-s} a^{k} \mathcal{C}\_{r}^{\*k} \times \\ \tag{59}$$
 
$$\times E\_{ll} \left[ f\_{Y}(F^{-1}(ll))^{q-1} lI^{k-s} | lI > F\_{Y}(t) \right] \right\}\_{l}$$

*where U is an U*(0, 1) *random variable and EU is the conditional expectation of fY*(*F*−1(*U*))*q*−1*Uk*−*<sup>s</sup> , given U* > *FY*(*t*)*.*

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

$$\overline{H}^{T}(Y\_{[r]}^{\*};t) = \frac{1}{q-1} \left\{ 1 - \frac{1}{\left(\mathcal{G}\_{[r]}(t)\right)^{q}} \sum\_{k=0}^{q} \sum\_{s=0}^{k} (-1)^{s} \binom{q}{k} \binom{k}{s} 2^{k-s} a^{k} \mathcal{C}\_{r}^{\*k} \times \\ \tag{60}$$
 
$$\times E\_{\mathcal{U}} \left[ f\_{Y}(F\_{Y}^{-1}(\mathcal{U}))^{q-1} \mathcal{U}^{k-s} | \mathcal{U} < F\_{Y}(t) \right] \right\},$$

*where U is a U*(0, 1) *random variable and EU is the conditional expectation of fY*(*F*−1(*U*))*q*−1*Uk*−*<sup>s</sup> , given U* < *FY*(*t*)*.*

**Corollary 4.** *The residual Tsallis entropy for the concomitant of r-th-order statistic is:*

$$H^{T}(Y\_{[r]};t) = \frac{1}{q-1} \left\{ 1 - \frac{1}{\left(\mathcal{G}\_{[r]}(t)\right)^{q}} \sum\_{k=0}^{q} \sum\_{s=0}^{k} (-1)^{s} \binom{q}{k} \binom{k}{s} 2^{k-s} a^{k} (-1)^{k} \left(\frac{n-2r+1}{n+1}\right)^{k} \times \\ \qquad \text{(61)}$$

$$\times E\_{II} \left[ f\_{Y}(F^{-1}(tI))^{q-1} L^{k-s} |II > F\_{Y}(t) \right] \right\},$$

*The residual Tsallis entropy for the concomitant of r-th record value is:*

$$H^{\mathbb{T}}(R\_{[r]};t) = \frac{1}{q-1} \left\{ 1 - \frac{1}{\left(\mathcal{G}\_{[r]}(t)\right)^q} \sum\_{k=0}^q \sum\_{s=0}^k (-1)^s \binom{q}{k} \binom{k}{s} 2^{k-s} a^k (-1)^k \left(2^{1-r} - 1\right)^k \times \tag{62}$$

$$\times E\_{\mathcal{U}} \Big[ f\_Y(F^{-1}(\mathcal{U}))^{q-1} \mathcal{U}^{k-s} | \mathcal{U} > F\_Y(t) \Big] \Big\}.$$

*In the case of progressive type II censoring order statistics with equi-balanced censoring scheme, the residual Tsallis entropy of the concomitant of r-th-order statistic from the FGM family is* (59) *with m* = *R, the removal number.*

*Similar results can be obtained for past Tsallis entropy.*

We now introduce **Residual Tsallis–Awad entropy**:

$$H^{TA}(X;t) = \frac{1}{q-1} \left\{ 1 - \frac{1}{\left(\delta^{\overline{F}}\_{(t,+\infty)}\right)^{q-1}} \int\_{t}^{+\infty} \left[\frac{f(x)}{\overline{F}(t)}\right]^{q} dx \right\} \tag{63}$$

$$I = \frac{1}{q-1} \left\{ 1 - \frac{1}{\overline{F}(t)} E\left[ \left. \frac{f(X)}{\overline{F}(t) \delta\_{(t,\infty)}^{\overline{T}}} \right|^{q-1} \right| X > t \right] \right\},\tag{64}$$

where *δ<sup>F</sup>* (*t*,∞) <sup>=</sup> <sup>1</sup> *<sup>F</sup>*(*t*) sup{ *<sup>f</sup>*(*x*)|*<sup>x</sup>* ∈ (*t*, +∞)}.

**Past Tsallis–Awad entropy** can also be defined:

$$\mathbf{T}^{\rm TA}(\mathbf{X};t) = \frac{1}{q-1} \left[ 1 - \frac{1}{\left(\delta^{\rm F}\_{(0,t)}\right)^{q-1}} \int\_0^t \left[\frac{f(\mathbf{x})}{F(t)}\right]^q d\mathbf{x} \right] \tag{65}$$

$$\hat{\lambda} = \frac{1}{q-1} \left\{ 1 - \frac{1}{F(t)} E\left[ \left[ \frac{f(X)}{F(t)\delta\_{(0,t)}^F} \right]^{q-1} \Big| 0 < X < t \right] \right\}.\tag{66}$$

where *δ<sup>F</sup>* (0,*t*) <sup>=</sup> <sup>1</sup> *<sup>F</sup>*(*t*) sup{ *f*(*x*)|*x* ∈ (0, *t*)}.

We notice that a similar relationship to (54) can be written for residual Tsallis entropies and for past Tsallis entropies:

$$H^{TA}(X;t) = \left(\delta^{\overline{F}}\_{(t,\infty)}\right)^{1-q} H^{T}(X;t) + \log\_q \delta^{\overline{F}}\_{(t,\infty)'}\tag{67}$$

$$
\overline{H}^{TA}(X;t) = \left(\delta^{F}\_{(0,t)}\right)^{1-q} \overline{H}^{T}(X;t) + \log\_{q} \delta^{F}\_{(0,t)'} \tag{68}
$$

and the following theorem can be proven:

**Theorem 5.** *Residual Tsallis–Awad entropy for the concomitant of the r-th GOS from the FGM family is:*

$$H^{TA}(Y\_{[r]}^{\*};t) = \frac{(\boldsymbol{\delta}^{G\_{[r]}}\_{(t,\infty)})^{1-q}}{q-1} \left\{ 1 - \frac{1}{\left(\overline{\mathbf{C}}\_{[r]}(t)\right)^{q}} \sum\_{k=0}^{q} \sum\_{s=0}^{k} (-1)^{s} \binom{q}{k} \binom{k}{s} 2^{k-s} a^{k} \mathbf{C}\_{r}^{\*k} \times \qquad \text{(69)}$$

$$\times \boldsymbol{E}\_{\rm II} \left[ f\_{Y}(\boldsymbol{F}\_{Y}^{-1}(\boldsymbol{U}))^{q-1} \boldsymbol{U}^{k-s} | \boldsymbol{U} > \boldsymbol{F}\_{Y}(t) \right] \right\} + \log\_{q} \boldsymbol{\delta}^{\overline{\mathbf{C}}\_{[r]}}\_{(t,\infty)}$$

*where U is an U*(0, 1) *random variable, and δ G*[*r*] (*t*,∞) <sup>=</sup> <sup>1</sup> *<sup>G</sup>*[*r*](*t*) sup{*g*[*r*](*x*)|*<sup>x</sup>* <sup>∈</sup> (*t*, <sup>+</sup>∞)}*. Past Tsallis–Awad entropy for the concomitant of the r-th GOS from the FGM family is:*

$$\overline{H}^{\rm TA}(Y\_{[r]}^{\*};t) = \frac{(\delta^{\rm G}\_{(0,t)})^{1-q}}{q-1} \left\{ 1 - \frac{1}{\left(\mathcal{G}\_{[r]}(t)\right)^{q}} \sum\_{k=0}^{q} \sum\_{s=0}^{k} (-1)^{s} \binom{q}{k} \binom{k}{s} 2^{k-s} a^{k} \mathcal{C}\_{r}^{\*k} \times \qquad \text{(70)}$$

$$\times E\_{\rm II} \left[ f\_{Y}(F\_{Y}^{-1}(\mathcal{U}))^{q-1} \mathcal{U}^{k-s} | \mathcal{U} < F\_{Y}(t) \right] \right\} + \log\_{q} \delta^{\rm G\_{[r]}}\_{(0,t)'}$$

*where U is an U*(0, 1) *random variable, and δ G*[*r*] (0,*t*) <sup>=</sup> <sup>1</sup> *G*[*r*] sup{*g*[*r*](*x*)|*x* ∈ (0, *t*)}*.*

**Corollary 5.** *The residual Tsallis–Awad entropy for the concomitant of r-th-order statistic is:*

$$H^{TA}(Y\_{[r]};t) = \frac{(\delta^{\Xi\_{[r]}}\_{(t,\infty)})^{1-q}}{q-1} \left\{ 1 - \frac{1}{\left(\overline{\mathbf{G}}\_{[r]}(t)\right)^q} \sum\_{k=0}^q \sum\_{s=0}^k (-1)^s \binom{q}{k} \binom{k}{s} 2^{k-s} a^k (-1)^k \left(\frac{n-2r+1}{n+1}\right)^k \times \\ \tag{71}$$
 
$$\times E\_{ll} \left[ \left\{ \mathbf{y}\_I(F\_Y^{-1}(lL))^{q-1} lI^{k-s} | lI > F\_Y(t) \right\} \right] + \log\_q s \frac{\mathcal{Z}\_{[r]}}{(t,\infty)}.$$

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

$$\begin{split} H^{\operatorname{TA}}(Y\_{[r]};t) = \frac{(\delta^{\operatorname{G}\_{[r]}}\_{(t,\infty)})^{1-q}}{q-1} \Bigg\{ 1 - \frac{1}{\left(\operatorname{\mathbf{G}}\_{[r]}(t)\right)^{q}} \sum\_{k=0}^{q} \sum\_{s=0}^{k} (-1)^{s} \binom{q}{k} \binom{k}{s} 2^{k-s} a^{k} (-1)^{k} \left(2^{1-r}-1\right)^{k} \times \\ & \qquad \times \operatorname{\mathbf{E}}\_{\mathcal{U}} \Big[ \bar{f} \circ (F\_{Y}^{-1}(\mathcal{U}))^{q-1} \mathcal{U}^{k-s} | \mathcal{U} > F\_{Y}(t) \Big] \Bigg\} + \log\_{q} \delta^{\operatorname{\mathbf{E}}\_{[r]}}\_{(t,\infty)}. \end{split} \tag{72}$$

*In the case of progressive type II censoring order statistics with equi-balanced censoring scheme, the residual Tsallis entropy of the concomitant of r-th-order statistic from the FGM family is* (59) *with m* = *R, the removal number.*

*Similar results can be obtained for past Tsallis entropy.*

#### *4.3. Fisher–Tsallis Information Number*

Various generalizations of FIN have been proposed, see, for example, [53–55]. In [53], the FIN is generalized, replacing the expectation and the logarithm functions with their *q* variants, and in [54], a (*β*, *q*)-Fisher information is defined. We here consider the following extension FIN, which we call **Fisher–Tsallis information number**:

$$I\_f = E\_f \left[ \left( \frac{\partial}{\partial \boldsymbol{\omega}} \log\_q f(\boldsymbol{\omega}) \right)^2 \right],\tag{73}$$

where log*<sup>q</sup>* is given by (30). This extension is a type of extension from [54], with *β* = 2 and *q* = 1.

For the concomitants of the GOS from FGM family, we have the following theorem which can be seen as an extension of the results obtained in [40]:

**Theorem 6.** *For the r-th concomitant Y*∗ [*r*] *of GOS from a FGM family, the Tsallis–Fisher information number for a location parameter is:*

$$I\_{\mathcal{S}\_{[r]}} = I\_1 + I\_2 + I\_{\mathcal{S}\_{\prime}} \tag{74}$$

*where*

$$\begin{split} I\_{1} &= E\_{lI} \Big[ \big( f\_{Y}(F^{-1}(\mathcal{U})) \big)^{-2q} \big( f\_{Y}^{\prime}(F\_{Y}^{-1}(\mathcal{U})) \big)^{2} \big( 1 + \mathcal{C}\_{r}^{\*}a(1-2\mathcal{U}) \big)^{2-2q} \Big], \\ I\_{2} &= 4\mathcal{C}\_{r}^{\*2}a^{2} \cdot \operatorname{E}\_{lI} \Big[ \big( f\_{Y}(F^{-1}(\mathcal{U})) \big)^{4-2q} (1+\mathcal{C}\_{r}^{\*}a(1-2\mathcal{U}))^{-2q} \Big], \\ I\_{3} &= -4\mathcal{C}\_{r}^{\*2}a \cdot \operatorname{E}\_{lI} \Big[ f\_{Y}(F^{-1}(\mathcal{U})) \big)^{2-2q} f\_{Y}^{\prime}(F\_{Y}^{-1}(\mathcal{U})) (1+\mathcal{C}\_{r}^{\*}a(1-2\mathcal{U}))^{1-2q} \Big]. \end{split}$$

**Proof.** From (73), we have

$$I\_{\mathcal{S}[r]} = E\_{\mathcal{S}[r]} \Big[ (\mathcal{g}\_{[r]}(\mathcal{Y}))^{-2\eta} (\mathcal{g}'\_{[r]}(\mathcal{Y}))^2 \Big].$$

Using the expression (7) for the density *g*[*r*], it results in

$$I\_{\mathcal{S}[r]} = E\_{\mathfrak{g}[r]} \left[ f\_Y(Y)^{-2q} \left[ 1 + \mathbb{C}\_r^\* a(1 - 2F\_Y(Y)) \right]^{-2q} \left[ f\_Y'(Y)(1 + \mathbb{C}\_r^\* a(1 - 2F\_Y(Y))) - 2(f\_Y(Y))^2 \mathbb{C}\_r^\* a \right]^2 \right].$$

Thus,

$$\begin{split} I\_{\mathcal{S}[r]} &= E\_{\mathcal{S}[r]} \left[ (f\_Y(Y))^{-2q} \left[ 1 + \mathcal{C}\_r^\* a (1 - 2F\_Y(Y)) \right] \right]^{-2q} \\ &\cdot \left[ (f\_Y'(Y))^2 (1 + \mathcal{C}\_r^\* a (1 - 2F\_Y(Y)))^2 + 4 (f\_Y(Y))^4 \mathcal{C}\_r^{\*2} a^2 - 4 \right] \\ &- 4 f\_Y'(Y) (f\_Y(Y))^2 \mathcal{C}\_r^\* a (1 + \mathcal{C}\_r^\* a (1 - 2F\_Y(Y))) \right]. \end{split}$$

After some computations,

$$\begin{split} I\_{\mathbb{S}[r]} &= \mathbb{E}\_{\mathbb{S}[r]} \left[ \left( f\_{Y}(Y) \right)^{-2q} (f\_{Y}^{\prime}(Y))^{2} \left( 1 + \mathsf{C}\_{r}^{\*} a \left( 1 - 2 \mathsf{F}\_{Y}(Y) \right) \right)^{2 - 2q} \right] + \\ &+ 4 \mathsf{C}\_{r}^{\* \* 2} a^{2} \mathbb{E}\_{\mathbb{S}[r]} \left[ f\_{Y}(Y)^{4 - 2q} \left( 1 + \mathsf{C}\_{r}^{\*} a \left( 1 - 2 \mathsf{F}\_{Y}(Y) \right) \right)^{-2q} \right] - \\ &- 4 \mathsf{C}\_{r}^{\*} a \mathbb{E}\_{\mathbb{S}[r]} \left[ \left( f\_{Y}(Y) \right)^{2 - 2q} f\_{Y}^{\prime}(Y) \left( 1 + \mathsf{C}\_{r}^{\*} a \left( 1 - 2 \mathsf{F}\_{Y}(Y) \right) \right)^{1 - 2q} \right]. \end{split}$$

After the transformation *U* = *FY*(*Y*), with *U* ∼ *U*(0, 1) we obtain (74).
