2. Connection between entropy and GMD

Let $X$ be a non-negative random variable with absolute continuous distribution function $F$ and let $\bar {F}(x)=P(X \gt x)$ denote the survival function. Let us further assume that the mean $\mu =E(X) \lt \infty$.

2.1. Cumulative residual entropy and GMD

We now consider the generalized cumulative residual entropy measure introduced recently by Sudheesh et al. [Reference Sudheesh, Sreedevi and Balakrishnan29].

Definition 1. Let $X$ be a non-negative random variable with absolute continuous distribution function $F$. Furthermore, let $\phi (\cdot)$ be a function of $X$ and $w(\cdot )$ be a weight function. Then, the generalized cumulative residual entropy is defined as

(1) \begin{equation} \mathcal{GE}(X)=\int_{0}^{\infty} w(u) E[\phi(X)-\phi(u)\,|\, X \gt u]\,dF(u). \end{equation}

Sudheesh et al. [Reference Sudheesh, Sreedevi and Balakrishnan29] showed that several measures of entropy available in the literature can be deduced from (1) with different choices of $w(\cdot )$ and $\phi (\cdot )$. In particular, for the choice of $w(u) = 1$ and $\phi (x) = x$, (1) reduces to the cumulative residual entropy [Reference Asadi and Zohrevand3].

We now show that GMD is a special case of $\mathcal {GE}(X)$.

Theorem 1. For the choice of $w(u)=\bar {F}(u)$ and $\phi (x)=2x$, $\mathcal {GE}(X)={\rm GMD}$.

Proof. Let $X_1$ and $X_2$ be independent and identical random variables having the distribution function $F$. With ${\rm GMD}=E|X_1-X_2|$, and because $|X_1-X_2|=\max (X_1,X_2)-\min (X_1,X_2)$, the GMD can alternatively be expressed as

$${\rm GMD}=E(\max(X_1,X_2)-\min(X_1,X_2)).$$

Now, for the choice of $w(u)=\bar {F}(u)$ and $\phi (x)=2x$, the expression in (1) becomes

\begin{align*} \mathcal{GE}(X)& =2\int_{0}^{\infty}\bar{F}(u)E(X-u\,|\,X \gt u)\,dF(u)\\ & =2\int_{0}^{\infty}\bar{F}(u)\left(\frac{1}{\bar{F}(u)}\int_{u}^{\infty}(y-u)\,dF(y)\right)dF(u)\\ & =2\int_{0}^{\infty}\int_{u}^{\infty}y \,dF(y)\,dF(u)-2\int_{0}^{\infty}u\int_{u}^{\infty}dF(y)\,dF(u) \\ & =\int_{0}^{\infty}y\cdot2F(y)\,dF(y)-\int_{0}^{\infty}y\cdot2\bar{F}(y)\,dF(y)\\ & =E(\max(X_1,X_2))-E(\min(X_1,X_2)), \end{align*}

which is precisely the GMD.

Remark 1. In general, for the choices of $w(u)=\bar {F}(u)$ and $\phi (x)=2x^v$ for some positive integer $v$, we have

$$\mathcal{GE}(X)=E(\max(X_1,X_2)^v-\min(X_1,X_2)^v).$$

Also, for $w(x)=\bar {F}^{k-1}(u)$, $k=2,3,\ldots$ and $\phi (x)=kx^v$, $v =1,2,\ldots$, we get

$$\mathcal{GE}(X)=E(\max(X_1,\ldots,X_{k})^v-\min(X_1,\ldots,X_{k})^v).$$

In the particular case of $v=1$, we have

\begin{align*} \mathcal{GE}(X)& =E(\max(X_1,\ldots,X_{k})-\min(X_1,\ldots,X_{k})) \\ & =E(\max(X_1,\ldots,X_{k}))-E(X_1)+E(X_1)-(E(\min(X_1,\ldots,X_{k})) \\ & =EG_k({-}X)-EG_k(X), \end{align*}

where $EG_k(X)$ and $EG_k(-X)$ are the risk-premium and the gain-premium of $X$ of order $k$, respectively. These two quantities have found key applications in bid and ask prices in finance; see Agouram and Lakhnati [Reference Agouram and Lakhnati2]. The above expression can alternatively be expressed as

\begin{align*} \mathcal{GE}(X)& = E(\max(X_1,\ldots,X_{k}))-E(X_1)+E(X_1)-(E(\min(X_1,\ldots,X_{k})) \\ & =k\,{\rm Cov}(X,F^{k-1}(X))-k\,{\rm Cov}(X,\bar F^{k-1}(X)). \end{align*}

In particular, when $k=2$, we obtain

$$\mathcal{GE}(X)=4{\rm Cov}(X,F(X)),$$

which is yet another representation of GMD [Reference Lerman and Yitzhaki17].

2.2. Generalized cumulative past entropy and GMD

We now consider the generalized cumulative past entropy and discuss its relationship with GMD.

Definition 2. [Reference Sudheesh, Sreedevi and Balakrishnan29]

Let $X$ be a non-negative random variable with absolute continuous distribution function $F$. Furthermore, let $\phi (\cdot )$ be a function of $X$ and $w(\cdot )$ be a weight function. Then, the generalized cumulative past entropy is defined as

(2) \begin{equation} \mathcal{GCE}(X)=\int_{0}^{\infty}w(u)E[\phi(u)-\phi(X)\,|\,X\le u]\,dF(u). \end{equation}

In particular, for the choice of $w(u)=1$ and $\phi (x)=x$, (2) reduces to $\mathcal {CE} (X)$ [Reference Di Crescenzo and Longobardi12].

Theorem 2. For the choice of $w(u)=F(u)$ and $\phi (x)=x$, $\mathcal {GCE}(X)$ reduces to the GMD.

Proof. Consider

\begin{align*} \mathcal{GE}(X)& =2\int_{0}^{\infty}{F}(u)E(u-X\,|\,X \lt u)\,dF(u)\\ & =2\int_{0}^{\infty}{F}(u)\left(\frac{1}{{F}(u)}\int_{0}^{u}(u-y)\,dF(y)\right)dF(u)\\ & =2\int_{0}^{\infty}\int_{0}^{u}u\,dF(y)\,dF(u)-2\int_{0}^{\infty}\int_{0}^{u}y\,dF(y)\,dF(u) \\ & =E(\max(X_1,X_2))-\int_{0}^{\infty}y\cdot 2\bar{F}(y)\,dF(y)\\ & =E(\max(X_1,X_2)-\min(X_1,X_2)), \end{align*}

as required.

Furthermore, for the choice of $w(u)=F^k(u)$ and $\phi (x)=(k+1)x$, where $k$ is a positive integer, we have

$$\mathcal{GCE}(X)=E(\max(X_1,\ldots,X_{k+1})-\min(X_1,\ldots,X_{k+1})).$$

We now show that GMD is a special case of the cumulative residual Tsallis entropy of order $\alpha$ defined by Rajesh and Sunoj [Reference Rajesh and Sunoj23] as

$${\rm CRT}_{\alpha}(X)=\frac{1}{\alpha-1}\int_{0}^{\infty} (\bar{F}(x)-\bar F^{\alpha}(x))\,dx, \quad \alpha \gt 0,\ \alpha\ne 1.$$

For $\alpha =2$, the above expression becomes

$${\rm CRT}_{2}(X)=\int_{0}^{\infty} (\bar{F}(x)-\bar F^{2}(x))\,dx.$$

For a non-negative random variable $X$, we have $\mu =E(X)=\int _{0}^{\infty }\bar F(x)\,dx$. Noting that $\bar F^{2}(x)$ is the survival function of $\min (X_1,X_2)$, we simply obtain

$${\rm CRT}_{2}(X)=E(X_1)-E(\min(X_1,X_2)).$$

Now, because $|X_1-X_2|=X_1+X_2-2\min (X_1,X_2)$, we obtain ${\rm GMD}=2 {\rm CRT}_{2}(X)$.

3. Connection between extropy and GMD

Tahmasebi and Toomaj [Reference Tahmasebi and Toomaj30] discussed a relationship between GMD and cumulative past extropy. Now upon using the generalized entropy measures introduced by Sudheesh et al. [Reference Sudheesh, Sreedevi and Balakrishnan29], we establish here some relationships between different dynamic extropy measures and the right- and left-truncated GMD.

First, we express the GMD in terms of cumulative residual extropy and cumulative past extropy. For a non-negative random variable $X$, Jahanshahi et al. [Reference Jahanshahi, Zarei and Khammar14] defined the cumulative residual extropy as

(3) \begin{equation} \mathcal{CRJ}(X)={-}\frac{1}{2}\int_{0}^{\infty}\bar{F}^2(x)\,dx, \end{equation}

while the cumulative past extropy is defined as Tahmasebi and Toomaj [Reference Tahmasebi and Toomaj30]

$$\mathcal{CJ}(X)={-}\frac{1}{2}\int_{0}^{\infty}(1-{F}^2(x))\,dx.$$

Various applications of $\mathcal {CRJ}(X)$ and $\mathcal {CJ}(X)$ have been discussed by Jahanshahi et al. [Reference Jahanshahi, Zarei and Khammar14] and Tahmasebi and Toomaj [Reference Tahmasebi and Toomaj30].

Using the survival functions of $\max (X_1,X_2)$ and $\min (X_1,X_2)$, we easily find that

\begin{align*} 2\mathcal{CRJ}(X)-2\mathcal{CJ}(X)& =\int_{0}^{\infty}(1-{F}^2(x))\,dx-\int_{0}^{\infty}\bar{F}^2(x)\,dx\\ & =E(\max(X_1,X_2)-\min(X_1,X_2))\\ & ={\rm GMD}. \end{align*}

Sudheesh and Sreedevi [Reference Sudheesh and Sreedevi28] discussed the non-parametric estimation of $\mathcal {CRJ}(X)$ and $\mathcal {CJ}(X)$ for right censored data, and so using the above relationship, we can readily obtain an estimator of the GMD based on right censored data.

Sathar and Nair [Reference Sathar and Nair25] defined dynamic survival extropy as

(4) \begin{equation} J_t(X)={-}\frac{1}{2\bar{F}^2(t)}\int_{t}^{\infty}\bar{F}^2(x)\,dx, \end{equation}

while Kundu [Reference Kundu15] introduced dynamic cumulative extropy as

$$H_t(X)={-}\frac{1}{2{F}^2(t)}\int_{0}^{t}{F}^2(x)\,dx.$$

Sudheesh and Sreedevi [Reference Sudheesh and Sreedevi28] proposed simple alternative expressions for different extropy measures, and then used them to establish some relationships between different dynamic and weighted extropy measures and reliability concepts.

The left-truncated GMD is given by

$${\rm GMD}_L=\frac{1}{\bar{F}^2(t)}\int_{t}^{\infty} (\bar{F}(t)-2\bar{F}(x))x\,dF(x)$$

while the right-truncated GMD is defined as

$${\rm GMD}_R=\frac{1}{{F}^2(t)}\int_{0}^{t} (2{F}(x)-{F}(t))x\,dF(x).$$

Proof. Consider the left-truncated GMD given by

(7) \begin{align} {\rm GMD}_L& =\frac{1}{\bar{F}^2(t)}\int_{t}^{\infty} (\bar{F}(t)-2\bar{F}(x))x\,dF(x)\nonumber\\ & =\frac{1}{\bar{F}^2(t)}\int_{t}^{\infty} (\bar{F}(t)-t+t-2\bar{F}(x))x\,dF(x)\nonumber\\ & =\frac{1}{\bar{F}(t)}\int_{t}^{\infty} (x-t)\,dF(x)-\frac{1}{\bar{F}^2(t)}\int_{t}^{\infty} (x-t)2\bar{F}(x)\,dF(x)\nonumber\\ & =m(t)-E(\min(X_1,X_2)-t\,|\min(X_1,X_2) \gt t). \end{align}

Sudheesh and Sreedevi [Reference Sudheesh and Sreedevi28] expressed $J_t(X)$ in (4) as

(8) \begin{equation} J_t(X)={-}\tfrac{1}{2}E(\min(X_1,X_2)-t\,|\min(X_1,X_2) \gt t). \end{equation}

Substituting (8) in (7), we obtain the relationship in (5).

Next, consider the right-truncated GMD given by

(9) \begin{equation} {\rm GMD}_R=\frac{1}{{F}^2(t)}\int_{0}^{t} (2{F}(x)-{F}(t))x\,dF(x). \end{equation}

After some algebraic manipulations, we obtain from (9) that

(10) \begin{equation} {\rm GMD}_R=E(t-\min(X_1,X_2)\,|\min(X_1,X_2)\le t)-r(x). \end{equation}

Sudheesh and Sreedevi [Reference Sudheesh and Sreedevi28] expressed $H_t(X)$ as

(11) \begin{equation} H_t(X)={-}\tfrac{1}{2}E(t-\max(X_1,X_2)\,|\max(X_1,X_2)\le t). \end{equation}

Substituting (11) in (10), we obtain the relationship in (6).

Proof. Using (1), we define the generalized residual entropy associated with $Z$ as

(12) \begin{equation} \mathcal{GE}(Z)=\int 2w(u) E[\phi(Z)-\phi(u)\,|\, Z \gt u]\bar{F}(u)\,dF(u), \end{equation}

where $\phi (\cdot )$ is a function of $Z$ and $w(\cdot )$ is a weight function. Now, for the choice of $w(u)=-\frac {1}{2}$ and $\phi (z)=z$, from (12), we obtain

$$\mathcal{GE}(Z)=\int ({\rm GMD}_L-m(u))\bar{F}(u)\,dF(u).$$

Proof. The generalized cumulative past entropy associated with $Z$ is defined as

(13) \begin{equation} \mathcal{GCE}(Z)=\int_{0}^{\infty}2w(u)E[\phi(u)-\phi(X)\,|\,X\le u]F(u)\,dF(u). \end{equation}

Again, for the choice of $w(u)=\frac {1}{2}$ and $\phi (z)=z$, from (13), we obtain

$$\mathcal{GCE}(Z)={-}\int_{0}^{\infty}({\rm GMD}_R+r(x))F(u)\,dF(u).$$

So, the generalized cumulative past entropy associated with $Z$ is the weighted average of sum of the right-truncated GMD and the mean past life.

4. Connections with PWMs

We now show that many of the measures discussed in the preceding sections can be represented in terms of PWM.

The PWM of a random variable $X$ with distribution function $F$ is defined as [Reference Greenwood, Landwehr, Matalas and Wallis13]

(14) \begin{equation} \mathcal{M}_{p,r,s}=\mathbb{E}\{ X^pF^r(X)(1-F(X))^s\}, \end{equation}

where $p$, $r$ and $s$ are any real numbers, for which the involved expectation exists.

First, we consider the GMD and related measures. From the proof of Theorem 1, we can express GMD as

$${\rm GMD}=\int_{0}^{\infty}2yF(y)\,dF(y)-\int_{0}^{\infty}2y\bar{F}(y)\,dF(y),$$

and so we have

$${\rm GMD}=2E(XF(X))-2E(X\bar{F}(X)),$$

which, when compared with (14), yields

$${\rm GMD}=2\mathcal{M}_{1,1,0}-2\mathcal{M}_{1,0,1}.$$

Several income inequality measures are derived from the GMD by choosing different weights in the expectation and one among them is the S-Gini family of indices [Reference Yitzhaki and Schechtman33]. The absolute S-Gini index is defined as

$$S_v={-}{\rm Cov}(X,\bar{F}^{v-1}(X)),\quad v \gt 0 \ \text{and}\ v\ne 1,$$

which can be rewritten as

$$S_v={-}E(X\bar{F}^{v-1}(X))+\frac{1}{v}E(X).$$

Hence, the absolute S-Gini index can be represented as

$$S_v=\frac{1}{v}\mathcal{M}_{1,0,0}-\mathcal{M}_{1,0,v-1}.$$

For $v=2$, $S_v$ reduces to GMD and hence GMD can also be represented in terms of PWM as

$${\rm GMD}=\frac{1}{v}\mathcal{M}_{1,0,0}-\mathcal{M}_{1,0,1}.$$

Next, we consider different extropy measures. The cumulative past extropy and the weighted cumulative past extropy of $X$ can be expressed as [Reference Sudheesh and Sreedevi28]

\begin{align*} \mathcal{CRJ}(X)& ={-}\tfrac{1}{2}E(\max(X_1,X_2)),\\ \mathcal{CRJW}(X)& ={-}\frac{1}{4}E(\max(X_1,X_2)^2). \end{align*}

Observing that $2F(x)f(x)$ is the density function of $\max (X_1,X_2)$, the extropy measures given above can be rewritten as

\begin{align*} \mathcal{CRJ}(X)& ={-}E(XF(X)),\\ \mathcal{CRJW}(X)& ={-}\tfrac{1}{2}E(X^2F(X)), \end{align*}

so that

$$\mathcal{CRJ}(X)={-}\mathcal{M}_{1,1,0}\quad \text{and}\quad \mathcal{CRJW}(X)={-}\tfrac{1}{2}\mathcal{M}_{2,1,0}.$$

The cumulative residual entropy and the weighted cumulative residual entropy of $X$ can be expressed as [Reference Sudheesh and Sreedevi28]

\begin{align*} \mathcal{CJ}(X)& ={-}\tfrac{1}{2}E(\min(X_1,X_2)),\\ \mathcal{WCJ}(X)& ={-}\tfrac{1}{4}E(\min(X_1,X_2)^2). \end{align*}

Here again, we observe that $2\bar {F}(x)f(x)$ is the density function of $\min (X_1,X_2)$, and so

\begin{align*} \mathcal{CJ}(X)& ={-}E(X\bar{F}(X))\\ \mathcal{WCJ}(X)& ={-}\tfrac{1}{2}E(X^2\bar{F}(X)), \end{align*}

from which we readily obtain

$$\mathcal{CJ}(X)={-}\mathcal{M}_{1,0,1}\quad \text{and}\quad \mathcal{WCJ}(X)={-}\tfrac{1}{2}\mathcal{M}_{2,0,1}.$$

Next, we show that the cumulative residual Tsallis entropy of order $\alpha$ can be written in terms of PWM. Let us assume that $\alpha$ is a positive integer with $\alpha \ne 1$. Recall that

(15) \begin{equation} {\rm CRT}_{\alpha}(X)=\frac{1}{\alpha-1}\int_{0}^{\infty} (\bar{F}(x)-\bar F^{\alpha}(x))\,dx. \end{equation}

We can show that

$$\int_{0}^{\infty}x \alpha \bar F^{\alpha-1}(x)\,dF(x) =\int_{0}^{\infty}\bar F^{\alpha}(x)\,dx,$$

and so we can write from (15) that

$${\rm CRT}_{\alpha}(X)=\frac{1}{\alpha-1}E(X)-\frac{\alpha}{\alpha-1}E(X\bar F^{\alpha}(X)),$$

which yields

$${\rm CRT}_{\alpha}(X)=\frac{1}{\alpha-1} \mathcal{M}_{1,0,0}-\frac{\alpha}{\alpha-1}\mathcal{M}_{1,0,(\alpha-1)}.$$

For a non-negative continuous random variable $X$, the weighted cumulative residual Tsallis entropy (WCRTE) of order $\alpha$ is defined as [Reference Chakraborty and Pradhan7]

$${\rm WCRT}_{\alpha}(X)=\frac{1}{\alpha-1}\int_{0}^{\infty}x(\bar{F}(x)-\bar{F}^{\alpha}(x))\,dx.$$

After some simple algebra, we can show that

$$\int_{0}^{\infty}x^2 \alpha \bar F^{\alpha-1}(x)\,dF(x) =\int_{0}^{\infty}2x\bar F^{\alpha}(x)\,dx,$$

so that

$${\rm WCRT}_{\alpha}(X)=\frac{1}{2(\alpha-1)}E(X^2)-\frac{\alpha}{2(\alpha-1)}E(X^2\bar{F}^{\alpha}(X))$$

from which we readily obtain

$${\rm WCRT}_{\alpha}(X)=\frac{1}{2(\alpha-1)} \mathcal{M}_{2,0,0}-\frac{\alpha}{2(\alpha-1)}\mathcal{M}_{2,0,(\alpha-1)}.$$

Calì et al. [Reference Calì, Longobardi and Ahmadi6] introduced the cumulative past Tsallis entropy of $X$ as

$$CT_{\alpha}(X)=\frac{1}{\alpha-1}\int_{0}^{\infty} ({F}(x)- F^{\alpha}(x))\,dx,\quad \alpha \gt 0, \ \alpha\ne1,$$

while Chakraborty and Pradhan [Reference Chakraborty and Pradhan8] defined the weighted cumulative past Tsallis entropy (WCTE) of order $\alpha$ as

$$WCT_{\alpha}(X)=\frac{1}{\alpha-1}\int_{0}^{\infty}x(F(x)-F^{\alpha}(x))\,dx,\quad \alpha \gt 0,\ \alpha\ne1.$$

As in the case of CRTE and WCRTE, we obtain

\begin{align*} CT_{\alpha}(X)& =\frac{1}{(\alpha-1)} \mathcal{M}_{1,0,0}-\frac{\alpha}{(\alpha-1)}\mathcal{M}_{1,(\alpha-1),0},\\ {\rm WCRT}_{\alpha}(X)& =\frac{1}{2(\alpha-1)} \mathcal{M}_{2,0,0}-\frac{\alpha}{2(\alpha-1)}\mathcal{M}_{2,(\alpha-1),0}. \end{align*}

Aaberge [Reference Aaberge1] introduced a family of inequality measures based on the moments of the Lorenz curve as

$$D_k(F)=\frac{1}{k\mu}\int_{0}^{\infty}F(x)(1-F^k(x))\,dx,\quad k=1,2,\ldots.$$

We note that $D_k(F)$ can be expressed in term of cumulative past Tsallis entropy as $CT_{k+1}(X)/\mu$. Hence, we can express

$$D_k(F)=\frac{1}{k\mu} \mathcal{M}_{1,0,0}-\frac{k+1}{k\mu}\mathcal{M}_{1,k,0}.$$

Sharma and Taneja [Reference Sharma and Taneja27] and Mittal [Reference Mittal20] independently introduced the STM entropy of $X$ as

(16) \begin{equation} S_{\alpha,\beta}=\frac{1}{\beta-\alpha}\int_{0}^{\infty}(f^{\alpha}(x)-f^{\beta}(x))\,dx. \end{equation}

Some entropy measures discussed in the literature can be derived from (16) by making different choices for $\alpha$ and $\beta$.

Sudheesh et al. [Reference Sudheesh, Sreedevi and Balakrishnan29] introduced a cumulative residual STM entropy of $X$ as

$$SR_{\alpha,\beta}=\frac{1}{\beta-\alpha}\int_{0}^{\infty}(\bar F^{\alpha}(x)-\bar F^{\beta}(x))\,dx,$$

and the cumulative past STM entropy of $X$ as

$$SP_{\alpha,\beta}=\frac{1}{\beta-\alpha}\int_{0}^{\infty}(F^{\alpha}(x)-F^{\beta}(x))\,dx.$$

They also introduced weighted versions of these measures as

$$SRW_{\alpha,\beta}=\frac{1}{\beta-\alpha}\int_{0}^{\infty}x(\bar F^{\alpha}(x)-\bar F^{\beta}(x))\,dx$$

and

$$SPW_{\alpha,\beta}=\frac{1}{\beta-\alpha}\int_{0}^{\infty}x(F^{\alpha}(x)-F^{\beta}(x))\,dx.$$

Sudheesh et al. [Reference Sudheesh, Sreedevi and Balakrishnan29] showed that $SR_{\alpha,\beta }$ and $SRW_{\alpha,\beta }$ are special cases of (1), while $SP_{\alpha,\beta }$ and $SPW_{\alpha,\beta }$ are special cases of (2).

We now express these measures in terms of PWM as follows, presented without proofs, for the sake of conciseness:

\begin{align*} SR_{\alpha,\beta}& =\frac{1}{\beta-\alpha}(\mathcal{M}_{1,0,(\alpha-1)}-\mathcal{M}_{1,0,(\beta-1)}),\\ SP_{\alpha,\beta}& =\frac{1}{\beta-\alpha}(\mathcal{M}_{1,(\alpha-1),0}-\mathcal{M}_{1,(\beta-1),0}),\\ SRW_{\alpha,\beta}& =\frac{1}{2(\beta-\alpha)}(\mathcal{M}_{2,0,(\alpha-1)}-\mathcal{M}_{2,0,(\beta-1)}),\\ SPW_{\alpha,\beta}& =\frac{1}{2(\beta-\alpha)}(\mathcal{M}_{2,(\alpha-1),0}-\mathcal{M}_{2,(\beta-1),0}). \end{align*}

These representations in terms of PWM would enable us to develop inferential methods for all these measures using the known results on PWM.

5. Concluding remarks

The GMD is a well-known measure of dispersion that is used extensively in economics. We have established here some relationships between the information measures and the GMD. We have shown that the GMD is a special case of the generalized cumulative residual entropy proposed recently by Sudheesh et al. [Reference Sudheesh, Sreedevi and Balakrishnan29]. We have also established some relationships between the GMD and extropy measures. We have further presented some relationships between the dynamic versions of the cumulative residual/past extropy and the truncated GMD.

The PWMs generalize the concept of moments of a probability distribution. The estimates derived using PWMs are often superior than the standard moment-based estimates. For discussion on different methods of estimation of the PWM, we refer to David and Nagaraja [Reference David and Nagaraja10], Vexler et al. [Reference Vexler, Zou and Hutson32] and Deepesh et al. [Reference Deepesh, Sudheesh and Sreelakshmi11] and the references therein. We have shown here that many of the information measures can be expressed in terms of PWM, which would facilitate the development of inferential methods for these measures using the well-known results on PWM.