# Weighted past and paired dynamic varentropy measures, their properties and
usefulness

S. Saha and S. Kayal
Dept. of Mathematics, NIT Rourkela, India Email address :
<EMAIL_ADDRESS>address (corresponding author) :
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

We introduce two uncertainty measures, say weighted past varentropy (WPVE) and
weighted paired dynamic varentropy (WPDVE). Several properties have been
studied for these proposed measures. The effect of the monotone transformation
for these measures have been discussed. We have obtained an upper bound of the
WPVE using the weighted past Shannon entropy. A lower bound of the WPVE is
also obtained. The WPVE has been studied for proportional reversed hazard rate
(PRHR) models. Upper and lower bounds of the WPDVE have been derived. We
propose non-parametric kernel estimates of the WPVE and WPDVE. Further,
maximum likelihood estimation has been employed to estimate WPVE and WPDVE for
an exponential population. A numerical simulation is provided to observe the
behaviour of the proposed estimates. Finally, we have analysed a real data set
and obtain the estimated values of WPVE.

Keywords: Weighted past varentropy, weighted paired dynamic varentropy,
monotone transformation, proportional reversed hazard rates model, non-
parametric estimate.

MSCs: 94A17; 60E15; 62B10.

##  1 Introduction

Consider a non-negative and absolutely continuous random variable (RV) $Y$.
Denote by $g(\cdot)$ the probability density function (PDF) of $Y.$ The
information content (IC) and weighted IC of $Y$ are

$\displaystyle
I(Y)=-\log\big{(}g(Y)\big{)}~{}~{}\text{and}~{}~{}I^{\omega}(Y)=-\omega(Y)\log\big{(}g(Y)\big{)},$
(1.1)

respectively, where $\omega(.)>0$ is called the weight function and ‘log’ is a
natural logarithm. In literature, $I(Y)$ and $I^{\omega}(Y)$ are also dubbed
as the Shannon IC and weighted Shannon IC, respectively. The rationale behind
$I(Y)$ can be provided in a discrete scenario, where it signifies the quantity
of bits that are fundamentally needed to represent $Y$ through a coding scheme
that reduces the average code length. For details, please refer to Shannon
(1948). The expectation of $I(Y)$, termed as Shannon entropy (SE) has been
widely studied by many authors (see for example, Hammer et al. (2000),
Kharazmi and Balakrishnan (2021), Saha et al. (2022)) in different fields of
research. The SE of $Y$, also known as differential entropy is defined as

$\displaystyle\mathcal{H}(Y)=E[I(Y)]=-\int_{0}^{\infty}g(y)\log\big{(}g(y)\big{)}dy.$
(1.2)

For a discrete RV $Y$, taking values $y_{i}$ with probabilities $p_{i}>0,$
$\sum_{i=1}^{n}p_{i}=1,$ $i=1,\dots,n$, the SE is

$\displaystyle\mathcal{H}(Y)=-\sum_{i=1}^{n}p_{i}\log(p_{i}).$ (1.3)

For details, please refer to Shannon (1948). Note that the SE measures
uncertainty or disorder contained in a RV $Y$. The amount of information and
entropy are inter-related. Higher entropy and disorder are correlated with
increased information; lower entropy and disorder are correlated with
decreased information. Clearly, (1.3) depends only on the probabilities of
occurrence of outcomes. Thus, (1.3) is not useful in many fields, dealing with
experiments where it is required to consider both probabilities and
qualitative characteristic of the events of interest. Thus, for distinguishing
the outcomes $y_{1},\dots,y_{n}$ of a goal-directed experiment according to
their importance with respect to a given qualitative characteristic of the
system, it is required to assign numbers $\omega_{k}>0$ to each outcome
$y_{k}$. One may choose $\omega_{k}$, proportional to the importance of the
kth outcome. Here, $\omega_{k}$’s are known as the weights of the outcomes
$y_{k},~{}k=1,\dots,n$. This type experiment is called as a weighted
probabilistic experiment. For such kind of experiments the weighted SE is
useful, which is defined as

$\displaystyle\mathcal{H}^{\omega}(Y)=-\sum_{i=1}^{n}\omega_{i}p_{i}\log(p_{i}).$
(1.4)

The continuous analogue of (1.4), known as the weighted SE or weighted
differential entropy of the RV $Y$ with weight $\omega(y)>0$ is defined as

$\displaystyle\mathcal{H}^{\omega}(Y)=-\int_{0}^{\infty}\omega(y)g(y)\log\big{(}g(y)\big{)}dy=E[I^{\omega}(Y)],$
(1.5)

For details, see Di Crescenzo and Longobardi (2006). Note that
$\mathcal{H}^{\omega}(X)$ in (1.5) is the expectation of weighted IC. It is a
measurement of the uncertainty and information provided by a probabilistic
experiment, which has been used to provide answers to many problems. The SE
and weighted SE are used in various field of areas such as computer science,
electrical engineering, behavioural science, environmental science, chemical
engineering and in coding theory (see Cover and Thomas (1991)). But there is a
discrimination that SE in (1.2) is a shift independent measure whenever the
weighted SE in (1.5) is shift dependent. As a result, weighted SE measure is
more flexible than SE.

Several researchers grow their interest to study the behaviour of the IC which
is useful in probability, statistics and information theory. The IC
concentrates around the SE in higher dimension with the log-concave PDF
function, which is studied by Bobkov and Madiman (2011). Occasionally, the SEs
of two RVs have the same value. For example, the SE of the exponential
distribution with rate parameter $e$ and uniform distribution in $(0,1)$ are
same. In this situation, the idea of the concentration of IC around SE is
helpful for analytical explanation. This concentration can be obtained as the
variance of $I(Y)$, which is known as the varentropy (VE). For a non-negative
absolutely continuous RV $Y$, the VE (see Fradelizi et al. (2016)) is
expressed as

$\displaystyle\mathcal{VE}(Y)=Var[I(Y)]=\int_{0}^{\infty}g(y)[\log\big{(}g(y)\big{)}]^{2}dy-\left[\mathcal{H}(Y)\right]^{2},$
(1.6)

where $\mathcal{H}(Y)$ is the SE of $Y$. Note that $\mathcal{VE}(Y)$
quantifies variability of $I(Y)$. For discrete RV $Y$, the VE is given by (see
Di Crescenzo and Paolillo (2021))

$\displaystyle\mathcal{VE}(Y)=\sum_{i=1}^{n}p_{i}[\log(p_{i})]^{2}-\left[\sum_{i=1}^{n}p_{i}\log(p_{i})\right]^{2}.$
(1.7)

One of the early appearances of the varentropy is when it was characterised as
the “minimal coding variance” studied by Kontoyiannis (1997). Further,
Kontoyiannis and Verdú (2014), used the concept of varentropy as “dispersion”
in source coding in computer science. Maadani et al. (2020) introduced
generalised varentropy based on Tsallis entropy and showed that the Tsallis
residual varentropy is independent of the age of the systems. Maadani et al.
(2022) proposed a method for calculating the varentropy for order statistics
and studied some stochastic comparison. Sharma and Kundu (2023) introduced the
concept of VE in a doubly truncated RV. The authors examined several
theoretical properties. Alizadeh Noughabi and Shafaei Noughabi (2023)
introduced some non-parametric estimates of the VE with some theoretical
properties. They compare the estimates based on the mean squared errors
(MSEs).

In survival analysis, the concept of the residual life is very useful for life
testing studies. Residual life-based informational measures are also useful
for predictive maintenance and decision-making in various fields like
reliability engineering, medicine science, and finance. Di Crescenzo and
Paolillo (2021) proposed residual varentropy (RVE) based on residual lifetime
of a system, $Y_{t}=[Y-t|Y>t]$, $t>0$ with PDF
$g_{t}(y)=\frac{g(y)}{\bar{G}(t)}$, where $\bar{G}(t)=P[Y>t]$ represents the
reliability function of $Y$. The RVE of $Y_{t}$ is defined as

$\displaystyle\mathcal{VE}(Y;t)=Var[I(Y_{t})]=\int_{t}^{\infty}\frac{g(y)}{\bar{G}(t)}\Big{(}\log\Big{(}\frac{g(y)}{\bar{G}(t)}\Big{)}\Big{)}^{2}dy-[\mathcal{H}(Y;t)]^{2},$
(1.8)

where $\mathcal{H}(Y;t)$ is the residual SE (see Ebrahimi and Pellerey
(1995)). They discussed several mathematical properties and provided two
applications pertaining to the first-passage timings of an Ornstein-Uhlenbeck
jump-diffusion process and the proportional hazards model.

The past lifetime occurs when we have failure before a specified inspection
time $t>0$. In many situations, it is necessary to measure uncertainty
contained in the past lifetime. For example, in forensic sciences and other
related fields, past lifetimes are used to analyse the right-censored data
(see Andersen et al. (2012)). Several researchers studied the uncertainty for
past lifetime in information theory. See, for instance Di Crescenzo and
Longobardi (2002), Di Crescenzo and Longobardi (2006), Di Crescenzo et al.
(2021), and Saha and Kayal (2023). Recently, Buono et al. (2022) introduced
varentropy of past lifetime. Let $G(t)=P[Y<t]$ be the cumulative distribution
function (CDF) of $Y$. The past lifetime of a system is denoted by
$Y^{*}_{t}=[t-Y|Y\leq t].$ The PDF of $Y^{*}_{t}$ is
$g^{*}_{t}(y)=\frac{g(y)}{G(t)}$. The VE of past lifetime is defined as

$\displaystyle\mathcal{VE^{*}}(Y;t)=\int_{0}^{t}\frac{g(y)}{G(t)}\Big{(}\log\Big{(}\frac{g(y)}{G(t)}\Big{)}\Big{)}^{2}dy-[\mathcal{H}^{*}(Y;t)]^{2},$
(1.9)

where $\mathcal{H}^{*}(Y;t)$ is the past SE (see Di Crescenzo and Longobardi
(2002)). Note that $\mathcal{VE^{*}}(Y;t)$ in (1.9) is the variance of the IC,
$I(Y^{*}_{t})=-\log\big{(}\frac{g(Y)}{G(t)}\big{)}$. Raqab et al. (2022)
considered past VE and obtained some reliability properties associated with
the past VE.

Very recently, Saha and Kayal (2024) proposed weighted varentropy (WVE) for
discrete as well as continuous RVs, and examined some properties. The authors
also studied WVE of the coherent systems. They further proposed weighted
residual varentropy (WRVE). The WVE is the variance of the weighted IC,
$I^{\omega}(Y)=-\omega(Y)\log(g(Y))$. Saha and Kayal (2024) showed that the
WVE gives better result than the VE for different distributions. For a
discrete RV $Y$, the WVE is given by

$\displaystyle\mathcal{VE}^{\omega}(Y)=\sum_{i=1}^{n}\omega^{2}_{i}p_{i}[\log(p_{i})]^{2}-\left[\sum_{i=1}^{n}\omega_{i}p_{i}\log(p_{i})\right]^{2},$
(1.10)

where $p_{i}$’s and $\omega_{i}$’s are probability mass function and weight
function corresponding to the event $Y=y_{i},$ for $i=1,\dots,n$. Analogously,
the WVE of $Y$ is defined as

$\displaystyle\mathcal{VE}^{\omega}(Y)=\int_{0}^{\infty}\omega^{2}(y)g(y)\Big{(}\log\big{(}g(y)\big{)}\Big{)}^{2}dy-[\mathcal{H}^{\omega}(Y)]^{2},$
(1.11)

where $\mathcal{H}^{\omega}(Y)$ is the weighted SE of $Y$ (see Di Crescenzo
and Longobardi (2006)). It is clear that VE as well as WVE are non-negative.
For uniform distribution, the value of VE is zero but WVE is non-zero. The
WRVE of $Y_{t}$ is defined as (see Saha and Kayal (2024))

$\displaystyle\mathcal{VE}^{\omega}(Y;t)=Var[IC^{\omega}(Y_{t})]=\int_{t}^{\infty}\frac{g(y)}{\bar{G}(t)}\bigg{(}\omega(y)\log\Big{(}\frac{g(y)}{\bar{G}(t)}\Big{)}\bigg{)}^{2}dy-[\mathcal{H}^{\omega}(Y;t)]^{2},$
(1.12)

where $\mathcal{H}^{w}(Y;t)$ is the weighted residual SE (see Di Crescenzo and
Longobardi (2006)) with weight $\omega(y)>0$. The authors have proposed non-
parametric estimator of the WRVE. Further, they have illustrated the proposed
estimate using a simulation study and two real data sets. In this
communication, motivated by the aforementioned findings and the usefulness of
the weight function in probabilistic experiment, we introduce weighted past
varentropy (WPVE) and studied its various properties. In the following, the
key contributions of this paper have been discussed.

  * •

In Section 2, we propose weighted varentropy for past lifetime. This measure
is called as the WPVE. The proposed measure is a generalisation of varentropy,
weighted varentropy and past varentropy. The WPVE has been studied under a
monotonically transformed RV. Lower and upper bounds of the WPVE are obtained.
Further, in Section 3 the WPVE has been studied for the PRHR model.

  * •

In Section 4, the concepts of weighted paired dynamic entropy (WPDE) and WPDVE
have been introduced. Several bounds of the WPDVE are obtained. The effect of
the WPDVE under an affine transformation has been examined.

  * •

In Section 5, the kernel-based non-parametric estimates of the WPVE and WPDVE
have been proposed. To see their performance, a Monte Carlo simulation study
has been carried out. For both WPVE and WPDVE, we have further considered
parametric estimation assuming that the data are taken from an exponential
population. Average daily wind speeds data set has been considered and
analysed. It is observed that the parametric estimates have superior
performance over the non-parametric estimates in terms of the absolute bias
(AB) and mean squared error (MSE) values.

Henceforth, we assume that the RVs are non-negative and absolutely continuous
unless it is mentioned. Further, ‘increasing’ and ‘decreasing’ are used in
wide sense. The differentiation, integration, and expectation always exist
wherever they are used.

##  2 Weighted past varentropy

In this section, we introduce an information measure by taking the variance of
the weighted IC for the past lifetime $Y^{*}_{t}$. The weighted IC of
$Y^{*}_{t}$ is $I^{\omega}(Y^{*}_{t})=-\omega(y)\log(\frac{g(y)}{G(t)}),$
where $\omega(y)>0$ is the weight function.

######  Definition 2.1.

Let $Y$ have the CDF $G(\cdot)$ and PDF $g(\cdot)$. The WPVE is defined by

$\displaystyle\mathcal{\overline{VE}^{\omega}}(Y;t)=Var[I^{\omega}(Y^{*}_{t})]=\int_{0}^{t}\frac{g(y)}{G(t)}\bigg{(}\omega(y)\log\Big{(}\frac{g(y)}{G(t)}\Big{)}\bigg{)}^{2}dy-[\overline{\mathcal{H}}^{\omega}(Y;t)]^{2},$
(2.1)

where $\overline{\mathcal{H}}^{\omega}(Y;t)$ is the weighted past SE (see Di
Crescenzo and Longobardi (2006)).

######  Remark 2.1.

The WPVE can be considered as a generalisation of the weighted VE (see Saha
and Kayal (2024)) and past VE (see Buono et al. (2022)). In particular, (2.1)
reduces to the past VE when $\omega(y)=1$, while (2.1) becomes the weighted VE
for $t\rightarrow\infty.$ Further, when $\omega(y)=1$ and
$t\rightarrow\infty$, then the WPVE coincides with the VE (Fradelizi et al.
(2016)).

For the weight function $\omega(y)=y,$ (2.1) can be written as

$\displaystyle\mathcal{\overline{VE}}^{y}(Y;t)=E[(\psi_{1}(Y))^{2}|Y\leq
t]-2\Lambda^{*}(t)\overline{\mathcal{H}}^{y^{2}}(Y;t)-(\Lambda^{*}(t))^{2}E[Y^{2}|Y\leq
t]-[\overline{\mathcal{H}}^{y}(Y;t)]^{2},$ (2.2)

where $\psi_{1}(y)=y\log(g(y))$. In (2.2), $\Lambda^{*}(t)=-\log(G(t))$ is the
cumulative reversed hazard rate (CRHR) function and
$\overline{\mathcal{H}}^{y^{2}}(Y;t)$ is the weighted past SE with weight
$y^{2}$. Now, we obtain the closed form expression of WPVE.

######  Example 2.1.

  * $(i)$

Suppose the uniform RV $Y$ has the CDF $G(y)=\frac{y-a}{b-a}$, $y\in[a,b]$.
The WPVE is

$\displaystyle\mathcal{\overline{VE}}^{y}(Y;t)=\frac{1}{12}\\{\log(t-a)\\}^{2}\\{4(t^{2}-at+a^{2})-3(t+a)^{2}\\},$

which is plotted in Figure 1$(a)$.

  * $(ii)$

For Pareto-I distribution with CDF $G(y)=1-y^{-\alpha},~{}y\geq
1,~{}\alpha>0$, the WPVE is obtained as

$\displaystyle\mathcal{\overline{VE}}^{y}(Y;t)=$
$\displaystyle\frac{\psi_{2}(t;\alpha)t^{2-\alpha}}{(2-\alpha)}\bigg{[}\left\\{\log\big{(}\psi_{2}(t;\alpha)\big{)}-(1+\alpha)\Big{(}\log(t)-\frac{1}{2-\alpha}\Big{)}\right\\}^{2}$
$\displaystyle+\frac{1}{2-\alpha}\left\\{2\log(t)-\frac{3-\alpha}{2-\alpha}\right\\}\bigg{]}-\big{(}\psi_{2}(t;\alpha)\big{)}^{2}\left(\frac{t^{1-\alpha}}{1-\alpha}\right)^{2}\bigg{[}\log\big{(}\psi_{2}(t;\alpha)\big{)}$
$\displaystyle-(1+\alpha)\left(\log(t)-\frac{1}{1-\alpha}\right)\bigg{]}^{2},$

where $\psi_{2}(t;\alpha)=\frac{\alpha}{1-t^{-\alpha}}.$ For graphical plot of
the WPVE of Pareto-I distribution, see Figure 1$(b)$.

  * $(iii)$

Consider an exponential RV $Y$ with CDF $G(y)=1-e^{-\lambda
y},~{}y>0,~{}\lambda>0$. The WPVE is

$\displaystyle\mathcal{\overline{VE}}^{y}(Y;t)=$ $\displaystyle
2\lambda\\{1-e^{-\lambda t}(1-\lambda
t)\\}\psi_{3}(t;\lambda)\left[\Big{\\{}\log\Big{(}\psi_{3}(t;\lambda)\Big{)}\Big{\\}}^{2}-6\log\Big{(}\psi_{3}(t;\lambda)\Big{)}+12\right]$
$\displaystyle-\frac{1}{\lambda^{2}(1-e^{-\lambda
t})^{2}}\left[\\{1-e^{-\lambda t}(1+\lambda
t)\\}\left\\{\log\big{(}\psi_{3}(t;\lambda)\big{)}-2\right\\}+\lambda^{2}t^{2}e^{-\lambda
t}\right]^{2}$ $\displaystyle-\frac{e^{-\lambda t}}{1-e^{-\lambda
t}}\bigg{[}t^{2}\Big{\\{}\log\Big{(}\psi_{3}(t;\lambda)\Big{)}\Big{\\}}^{2}+(6t^{2}+2t^{3})\log\Big{(}\psi_{3}(t;\lambda)\Big{)}$
$\displaystyle+(12t^{2}+4\lambda t^{3}+\lambda^{2}t^{4})\bigg{]},$

where $\psi_{3}(t;\lambda)=\frac{\lambda}{1-e^{-\lambda t}}.$ The plot of the
WPVE is provided in Figure 1$(c)$.

Figure 1: Graphs for the WPVE of $(a)$ uniform distribution in Example
2.1$(i)$, $(b)$ Pareto-I distribution in Example 2.1$(ii)$, and $(c)$
exponential distribution in Example 2.1$(iii)$.

Next, we obtain an upper bound of the WPVE via weighted past SE and CRHR
function.

######  Theorem 2.1.

Suppose $Y$ is a RV with PDF $g(\cdot).$ Further, let the PDF satisfy

$\displaystyle e^{-(\alpha y+\beta)}\leq g(y)\leq
1,~{}~{}y>0,~{}\alpha>0,~{}\beta\geq 0.$ (2.3)

Then, for $t>0$

$\displaystyle\mathcal{\overline{VE}}^{y}(Y;t)\leq\overline{\mathcal{H}}^{\omega_{2}}(Y;t)-2\Lambda^{*}(t)E[\alpha
Y^{3}+\beta Y^{2}|Y\leq t]+\Lambda^{*^{2}}(t)E[Y^{2}|X\leq t],$ (2.4)

where $\overline{\mathcal{H}}^{\omega_{2}}(Y;t)$ is the weighted past SE with
weight $\omega_{2}(y)=\alpha y^{3}+\beta y^{2}$.

###### Proof.

For $\omega(y)=y$, from (2.1) we get

$\displaystyle\mathcal{\overline{VE}}^{y}(Y;t)=\int_{0}^{t}\frac{g(y)}{G(t)}\bigg{(}y\log\left(\frac{g(y)}{G(t)}\right)\bigg{)}^{2}dy-[\overline{\mathcal{H}}^{y}(Y;t)]^{2}\leq\int_{0}^{t}y^{2}\frac{g(y)}{G(t)}\left[\log(g(y))+\Lambda^{*}(t)\right]^{2}dy.$
(2.5)

Further,

$\displaystyle\int_{0}^{t}y^{2}\frac{g(y)}{G(t)}[\log(g(y))+\Lambda^{*}(t)]^{2}dy$
$\displaystyle=\int_{0}^{t}y^{2}\frac{g(y)}{G(t)}[\log(g(y))]^{2}dy+2\int_{0}^{t}y^{2}\frac{g(y)}{G(t)}\log(g(y))\Lambda^{*}(t)dy$
$\displaystyle+\int_{0}^{t}y^{2}\frac{g(y)}{G(t)}[\Lambda^{*}(t)]^{2}dy$
$\displaystyle\leq-\int_{0}^{t}y^{2}(\alpha
y+\beta)\frac{g(y)}{G(t)}\log(g(y))dy+[\Lambda^{*}(t)]^{2}\int_{0}^{t}y^{2}\frac{g(y)}{G(t)}dy$
$\displaystyle-2\int_{0}^{t}y^{2}(\alpha
y+\beta)\frac{g(y)}{G(t)}\Lambda^{*}(t)dy$
$\displaystyle=\overline{\mathcal{H}}^{\omega_{2}}(Y;t)-2\Lambda^{*}(t)E[(\alpha
Y^{3}+\beta Y^{2})|Y\leq t]+\Lambda^{*^{2}}(t)E[Y^{2}|Y\leq t].$

∎

In the following example, we show that Lomax distribution satisfies the
condition in (2.3).

######  Example 2.2.

Consider the Lomax distribution with CDF
$G(x)=1-(1+\frac{x}{\delta})^{-\gamma},~{}x>0,~{}\delta>0,$ and $\gamma>0$.
The inequality in (2.3) can be easily checked from the graphical plot (see
Figure 2).

Figure 2: Graphical plots of $e^{-(\alpha x+\beta)}$ (blue colour) and
$g(x)=\frac{\gamma}{\delta}(1+x/\delta)^{-(\gamma+1)}$ (red colour) for
$\alpha=2,~{}\beta=1,~{}\delta=1,$ and $\gamma=3$. To capture the full support
$x\in(0,\infty)$, we take $x=-\log y$, where $y\in(0,1).$

For a RV $Y$, the variance past lifetime (VPL) is defined as

$\displaystyle\sigma^{2}(t)=Var[t-Y|Y\leq
t]=\frac{2}{G(t)}\int_{0}^{t}du\int_{u}^{t}g(z)dz-[\mathcal{M}(t)]^{2},$ (2.6)

where $\mathcal{M}(t)=\int_{0}^{t}\frac{G(y)}{G(t)}dy$ is called mean past
lifetime (MPL). For detailed study on VPL, please see Mahdy (2016). Now, we
obtain a lower bound of the WPVE in terms of the VPL.

######  Theorem 2.2.

We have

$\displaystyle\mathcal{\overline{VE}}^{\omega}(Y;t)\geq\sigma^{2}(t)\\{1+E[-\zeta_{t}(Y_{t})\log(g_{t}(Y_{t}))]+E[Y_{t}\zeta^{{}^{\prime}}_{t}(Y_{t})]\\}^{2},$
(2.7)

where $\zeta_{t}(\cdot)$ can be determined from

$\displaystyle\sigma^{2}(t)\zeta_{t}(y)g_{t}(y)=\int_{0}^{y}(\mathcal{M}(t)-u)g_{t}(u)du,~{}y>0.$
(2.8)

###### Proof.

Let $Y$ be a RV with PDF $g(\cdot)$, mean $m,$ and variance $\sigma^{2}$.
Then,

$\displaystyle
Var[\mathcal{I}(Y)]\geq\sigma^{2}(E[\eta(Y)\mathcal{I}^{{}^{\prime}}(Y)])^{2},$
(2.9)

where $\eta(.)$ can be obtained using
$\int_{0}^{y}(m-u)g(u)du=\sigma^{2}\eta(y)g(y)$ (see Cacoullos and
Papathanasiou (1989)). For proving the required result, we consider $Y_{t}$ as
a reference RV with $\mathcal{I}(y)=I^{y}(y)=-y\log(g(y))$. From (2.9), we get

$\displaystyle Var[-Y_{t}\log(g_{t}(Y_{t}))]$ $\displaystyle\geq$
$\displaystyle\sigma^{2}(t)\bigg{\\{}E\big{[}\zeta_{t}(Y_{t})\big{(}-Y_{t}\log(g_{t}(Y_{t}))\big{)}^{\prime}\big{]}\bigg{\\}}^{2}$
(2.10) $\displaystyle=$
$\displaystyle\sigma^{2}(t)\bigg{\\{}E[-\zeta_{t}(Y_{t})\log(g_{t}(Y_{t}))]-E\bigg{[}\zeta_{t}(Y_{t})Y_{t}\frac{g^{{}^{\prime}}_{t}(Y_{t})}{g_{t}(Y_{t})}\bigg{]}\bigg{\\}}^{2}.$

Further,

$\displaystyle
E\bigg{[}\zeta_{t}(Y_{t})Y_{t}\frac{g^{{}^{\prime}}_{t}(Y_{t})}{g_{t}(Y_{t})}\bigg{]}=E\bigg{[}Y_{t}\bigg{(}\frac{\mathcal{M}(t)-Y_{t}}{\sigma^{2}(t)}-\zeta^{{}^{\prime}}_{t}(Y_{t})\bigg{)}\bigg{]}=-1-E[Y_{t}\zeta^{{}^{\prime}}_{t}(Y_{t})].$
(2.11)

Using (2.11) in (2.10), the required result can be easily obtained. ∎

Next, we present a corollary. Its proof readily follows from Theorem 2.2.

######  Corollary 2.1.

  * (i)

Suppose $\zeta_{t}(y)$ is increasing function in $y>0$. Then,

$\overline{\mathcal{VE}}^{\omega}(Y;t)\geq\sigma^{2}(t)\big{(}E[\zeta_{t}(Y_{t})\log(g_{t}(Y_{t}))]\big{)}^{2}.$

  * (ii)

Let $g_{t}(y)\leq 1.$ Then,

$\overline{\mathcal{VE}}^{\omega}(Y;t)\geq\sigma^{2}(t)\big{(}E[Y_{t}\zeta^{{}^{\prime}}_{t}(Y_{t})]\big{)}^{2}.$

Sometimes, it is hard to evaluate the closed-form expression of the WPVE for a
transformed RV. The following theorem is useful to obtain the WPVE of a new
distribution constructed using a monotone transformation. Note that the
monotone transformations are useful tools that preserve entropy or information
of a RV in the field information theory.

######  Theorem 2.3.

Let $Y$ be a RV and $X=\psi(Y)$, where $\psi$ is a strictly monotonic,
continuous, and differentiable function. Then,

$\overline{\mathcal{VE}}^{x}(X;t)=\left\\{\begin{array}[]{ll}\overline{\mathcal{VE}}^{\psi}(Y;\psi^{-1}(t))-2\overline{\mathcal{H}}^{\psi}(Y;\psi^{-1}(t))E[\gamma_{1}(Y)|Y\leq\psi^{-1}(t)]\\\
+Var[\gamma_{1}(Y)|Y\leq\psi^{-1}(t)]-2E\Big{[}\psi(Y)\gamma_{1}(Y)\log\Big{(}\frac{g(Y)}{G(\psi^{-1}(t))}\Big{)}\Big{|}Y\leq\psi^{-1}(t)\Big{]},\\\
~{}if~{}\psi~{}is~{}strictly~{}increasing;\\\ \\\
{\mathcal{VE}}^{\psi}(Y;\psi^{-1}(t))-2\mathcal{H}^{\psi}(Y;\psi^{-1}(t))E[\gamma_{2}(Y)|Y>\psi^{-1}(t)]\\\
+Var[\gamma_{2}(Y)|Y>\psi^{-1}(t)]-2E\Big{[}\psi(Y)\gamma_{2}(Y)\log\Big{(}\frac{g(Y)}{G(\psi^{-1}(t))}\Big{)}\Big{|}Y>\psi^{-1}(t)\Big{]},\\\
~{}if~{}\psi~{}is~{}strictly~{}decreasing,\end{array}\right.$ (2.12)

where $\gamma_{1}(y)=\psi(y)\log(\psi^{\prime}(y))$,
$\gamma_{2}(y)=\psi(y)\log(-\psi^{\prime}(y))$,
$\mathcal{H}^{\psi}(Y;t)=-\int_{t}^{\infty}\psi(y)\frac{g(y)}{\bar{G}(t)}\log\big{(}\frac{g(y)}{\bar{G}(t)}\big{)}dy$,
$\mathcal{VE}^{\psi}(Y;t)=\int_{t}^{\infty}\psi^{2}(y)\frac{g(y)}{\bar{G}(t)}\left(\big{(}\frac{g(y)}{\bar{G}(t)}\big{)}\right)^{2}dy-[\mathcal{H}^{\psi}(Y;t)]^{2}$,
$\overline{\mathcal{H}}^{\psi}(Y;t)=-\int_{0}^{t}\psi(y)\frac{g(y)}{{G}(t)}\log\big{(}\frac{g(y)}{{G}(t)}\big{)}dy$,
and
$\overline{\mathcal{VE}}^{\psi}(Y;t)=\int_{0}^{t}\psi^{2}(y)\frac{g(y)}{{G}(t)}\Big{(}\log\big{(}\frac{g(y)}{{G}(t)}\big{)}\Big{)}^{2}dy-[\overline{\mathcal{H}}^{\psi}(Y;t)]^{2}$.

###### Proof.

Assume that $\psi$ is a strictly increasing function. After some calculations,
we obtain

$\displaystyle\overline{\mathcal{VE}}^{x}(X;t)$ $\displaystyle=$
$\displaystyle\int_{\psi^{-1}(0)}^{\psi^{-1}(t)}\psi^{2}(y)\frac{g(y)}{{G}(\psi^{-1}(t))}\bigg{[}\log\Big{(}\frac{g(y)}{{G}(\psi^{-1}(t))}\Big{)}\bigg{]}^{2}dy$
(2.13)
$\displaystyle-E[\psi^{2}(Y)\\{2\log(\psi^{{}^{\prime}}(Y))\log\Big{(}\frac{g(Y)}{G\big{(}\psi^{-1}(t)\big{)}}\Big{)}-(\log(\psi^{{}^{\prime}}(Y)))^{2}\\}|Y\leq\psi^{-1}(t)]$
$\displaystyle-\\{\mathcal{H}^{\psi}(Y;\psi^{-1}(t))+E[\psi(Y)\log(\psi^{{}^{\prime}}(Y))|Y\leq\psi^{-1}(t)]\\}^{2}.$

Now, the rest of the proof can be easily deduced from (2.13), and thus it is
skipped. The proof for strictly decreasing function $\psi$ is similar to that
when $\psi$ is strictly increasing. ∎

To see the validation of the result in Theorem 2.3, we consider the following
example.

######  Example 2.3.

Consider exponential RV $Y$ with the CDF $G_{1}(y)=1-e^{-\lambda y},~{}y>0$
and $\lambda>0$. Further, let $X=\psi(Y)=Y^{2}$, which is strictly increasing,
continuous, and differentiable function. Here, $X$ follows Weibull
distribution with CDF $G_{2}(y)=1-e^{-\lambda y^{\frac{1}{2}}},~{}y>0$ and
$\lambda>0$. Then, the WPVE of $X=\psi(Y)=Y^{2}$ is obtained as

$\displaystyle\overline{\mathcal{VE}}^{x}(X;t)$
$\displaystyle=\frac{\lambda}{1-e^{-\lambda\sqrt{t}}}\Big{[}t\sqrt{t}\Big{(}2t\log\Big{(}\frac{\lambda}{1-e^{-\lambda\sqrt{t}}}\Big{)}-\lambda\sqrt{t}-6\Big{)}e^{-\lambda\sqrt{t}}+\frac{1}{\lambda^{5}}\Big{\\{}24-\Big{(}24(1+\lambda\sqrt{t})$
$\displaystyle+12\lambda^{2}t+4\lambda^{3}t\sqrt{t}\lambda^{4}t^{2}\Big{)}e^{-\lambda\sqrt{t}}\Big{\\}}\Big{]}-\phi_{1}(\lambda;t)\Big{\\{}\phi_{1}(\lambda;t)+2E[Y^{2}\log(2Y)|Y\leq\sqrt{t}]\Big{\\}}$
$\displaystyle+\text{Var}[Y^{2}\log(2Y)|Y\leq\sqrt{t}]-2E\Big{[}Y^{4}\log(2Y)\log\Big{(}\frac{\lambda
e^{-\lambda Y}}{1-e^{-\lambda\sqrt{t}}}\Big{)}\Big{|}Y\leq\sqrt{t}\Big{]},$
(2.14)

where
$\phi_{1}(\lambda;t)=\frac{1}{\lambda^{2}(1-e^{-\lambda\sqrt{t}})}\Big{[}\\{2-(\lambda^{2}t+2\lambda\sqrt{t}+2)e^{-\lambda\sqrt{t}}\\}\log\Big{(}\frac{\lambda}{1-e^{-\lambda\sqrt{t}}}\Big{)}+(6+6\lambda\sqrt{t}+3\lambda^{2}t+\lambda^{3}t\sqrt{t})e^{-\lambda\sqrt{t}}-6\Big{]}$.
We have plotted the graphs of WPVE in (2.3) in Figure 3 with respect to $t$
(for fixed $\lambda$) and with respect to $\lambda$ (for fixed $t$).

Figure 3:  Plots of the WPVE for the Weibull distribution in Example 2.3 $(a)$
with respect to $t$ and $(b)$ with respect to $\lambda$.

Now, we investigate the effect of the WPVE under the affine transformation
$X=\alpha Y+\beta$, $\alpha>0$ and $\beta\geq 0$.

######  Corollary 2.2.

Suppose $X$ is a RV and $X=\alpha Y+\beta$ with $\alpha>0,~{}\beta\geq 0.$
Then,

$\displaystyle\overline{\mathcal{VE}}^{x}(X;t)=$
$\displaystyle\overline{\mathcal{VE}}^{\omega_{1}}\Big{(}Y;\frac{t-\beta}{\alpha}\Big{)}+\big{(}\log(\alpha)\big{)}^{2}\text{Var}\left[\alpha
Y+\beta\Big{|}Y\leq\frac{t-\beta}{\alpha}\right]$
$\displaystyle-2\log(a)\left(\overline{\mathcal{H}}^{\omega_{1}}\Big{(}Y;\frac{t-\beta}{\alpha}\Big{)}+\mathcal{\overline{H}}^{\omega_{1}}\Big{(}Y;\frac{t-\beta}{\alpha}\Big{)}E\bigg{[}\alpha
Y+\beta\Big{|}Y\leq\frac{t-\beta}{\alpha}\bigg{]}\right),$

where
$\overline{\mathcal{VE}}^{\omega_{1}}\left(Y;\frac{t-\beta}{\alpha}\right)$
and $\mathcal{\overline{H}}^{\omega_{1}}\left(Y;\frac{t-\beta}{\alpha}\right)$
are the WPVE and weighted past SE with weight $\omega_{1}(y)=\alpha y+\beta$,
respectively.

###### Proof.

We here omit the proof, since it readily follows from Theorem 2.3. ∎

Next, an example has been considered to illustrate Corollary 2.2.

######  Example 2.4.

Consider the CDF $G(y)=1-e^{-y},~{}y>0$. We take $X=Y+\beta,~{}\beta>0$. Thus,
from Corollary 2.2, the WPVE of $X$ is obtained as

$\displaystyle\overline{\mathcal{VE}}^{x}(X;t)$
$\displaystyle=\frac{1}{\psi_{3}(t;\beta)}\bigg{[}\Big{\\{}2\log(\psi_{3}(t;\beta))+(6+2\beta)e^{-\beta}\Big{\\}}\Big{\\{}\beta^{5}-(t-2\beta)^{5}e^{-(t-3\beta)}\Big{\\}}+\Big{\\{}\Big{(}\log\big{(}\psi_{3}(t;\beta)\big{)}\Big{)}^{2}$
$\displaystyle+2(5+\beta)\log(\psi_{3}(t;\beta))+5(6+2\beta)e^{-\beta}\Big{\\}}\Big{\\{}\beta^{4}+4(\beta^{2}-4\beta+6)e^{-\beta}-(t-2\beta)^{4}e^{-(t-3\beta)}$
$\displaystyle-(t-2\beta)^{4}e^{-(t-3\beta)}-4(3-\beta)e^{-(t-4\beta)}\Big{(}(t-2\beta+1)^{2}-(t-2\beta)^{3}\Big{)}\Big{\\}}\bigg{]}-\big{(}\psi_{4}(t;\beta)\big{)}^{2},$

where $\psi_{3}(t;\beta)=e^{-(t-\beta)}-1$,
$\psi_{4}(t;\beta)=\frac{1}{\psi_{3}(t;\beta)}\Big{[}(\beta^{2}-4\beta+6)-e^{-(t-3\beta)}\Big{\\{}(3-\beta)(t-2\beta+1)^{2}+(3-\beta)-(t-2\beta)^{3}\Big{\\}}+\log(\psi_{3}(t;\beta))\Big{\\{}(\beta^{2}-2\beta+2)+\big{(}(t-2\beta+1)^{2}+1\big{)}e^{-(t-3\beta)}\Big{\\}}\Big{]}$
and $\psi_{5}(t;\beta)=1-\beta+(2\beta-t-1)e^{-(t-\beta)}$. We have plotted
the graphs of the WPVE of $X$ in Figure 4. Clearly, the WPVE is non-monotone
with respect to $t$ and $\beta.$

Figure 4: Graphs of the WPVE in Example 2.4 $(a)$ with respect to $t$ (for
fixed $b$) and $(b)$ with respect to $\beta$ (for fixed $t$).

##  3 WPVE for PRHR model

The PRHR model is a crucial idea to use in reliability engineering, survival
analysis, industries, and various other fields. Gupta et al. (1998) introduced
PRHR model and discussed its properties. It offers flexibility in modeling the
reversed hazard rate of events that exhibit decreasing reversed failure rates
over time. This is particularly useful in situations where traditional models
(like the Weibull distribution) which assumes increasing or constant hazard
rates, may not adequately capture the behavior of the data. Let $Y$ and $X$ be
two RVs with CDFs $G_{1}(\cdot)$ and $G_{2}(\cdot)$, respectively. The PRHR
model of RVs $X$ and $Y$ is defined by

$\displaystyle G_{2}(t)=P[Y^{(a)}\leq t]=[G_{1}(t)]^{a},~{}~{}\text{for all
$t>0$ and $a>0$.}$ (3.1)

Its PDF is

$\displaystyle g_{2}(t)=a[G_{1}(t)]^{a-1}g_{1}(t),$ (3.2)

where $g_{1}(\cdot)$ and $g_{2}(\cdot)$ are the PDFs of $Y$ and $X$,
respectively. It is known as the Lehmans’s alternatives for $a>0$. Various
researchers studied PRHR model and introduced its several properties. Gupta
and Gupta (2007) discussed Fisher information in PRHR model. Li and Li (2008)
proposed a mixture model of PRHR model and discussed some properties. For some
properties of the PRHR model, we refer to Finkelstein (2002), Nanda and Das
(2011), Balakrishnan et al. (2018), and Popović et al. (2021). Suppose
$\tilde{\Lambda}(x)$ denotes the CRHR function of $X$, is defined by

$\displaystyle\tilde{\Lambda}(y)=-\log\big{(}{G_{2}}(y)\big{)}=-\log\big{(}[{G_{1}}(y)]^{a}\big{)}=a\Lambda^{*}(y),~{}y>0.$
(3.3)

The weighted past SE of $X$ is obtained as

$\displaystyle\overline{\mathcal{H}}^{y}(X;t)$ $\displaystyle=$
$\displaystyle-\int_{0}^{t}y\frac{g_{2}(y)}{{G_{2}}(t)}\log\big{(}g_{2}(y)\big{)}dy-\int_{0}^{t}y\frac{g_{2}(y)}{{G_{2}}(t)}\tilde{\Lambda}(t)dy$
(3.4) $\displaystyle=$
$\displaystyle-\frac{1}{[{G_{1}}(t)]^{a}}\bigg{\\{}\int_{0}^{[{G_{1}}(t)]^{a}}\mathcal{L}(x:a)dx+a\Lambda^{*}(t)\int_{0}^{[{G_{1}}(t)]^{a}}{G_{1}}^{-1}(x^{1/a})dx\bigg{\\}}$
$\displaystyle=$
$\displaystyle\frac{1}{[{G_{1}}(t)]^{a}}\int_{0}^{[{G_{1}}(t)]^{a}}\mathcal{J}(x:a,t)dx,$

where $x=[{G_{1}}(y)]^{a}$,
$\mathcal{L}(y:a)={G_{1}}^{-1}(x^{1/a})\log\big{(}ax^{1-1/a}g[{G_{1}}^{-1}(x^{1/a})]\big{)},$
and

$\displaystyle\mathcal{J}(x:a,t)=-{G_{1}}^{-1}(x^{1/a})\log\Big{(}a\frac{y^{1-1/a}}{[{G_{1}}(t)]^{a}}g_{1}[{G_{1}}^{-1}(x^{1/a})]\Big{)},$
(3.5)

where ${G_{1}}^{-1}(x^{1/a})=\sup\\{y:{G_{1}}(y)\leq x^{1/a}\\}$ is called the
quantile function of ${G_{1}(\cdot)}$. Next, we obtain WPVE for the PRHR model
in (3.1).

######  Theorem 3.1.

Suppose $X$ is a RV having CDF $G_{2}(\cdot)$ in (3.1) . Then, for $a>0$ and
$t>0$, the WPVE of $X$ is

$\displaystyle\overline{\mathcal{VE}}^{y}(X;t)$ $\displaystyle=$
$\displaystyle\frac{1}{[{G_{1}}(t)]^{a}}\int_{0}^{[{G_{1}}(t)]^{a}}[\mathcal{J}(x:a,t)]^{2}dx-\frac{1}{[{G_{1}}(t)]^{2a}}\bigg{\\{}\int_{0}^{[{G_{1}}(t)]^{a}}\mathcal{J}(x:a,t)dx\bigg{\\}}^{2},$
(3.6)

where $\mathcal{J}(x:a,t)$ is given in (3.5).

###### Proof.

Using (3.1) in (2.1), we have

$\displaystyle\overline{\mathcal{VE}}^{y}(X;t)=\int_{0}^{t}\frac{g_{2}(y)}{{G_{2}}(t)}\bigg{(}y\log\Big{(}\frac{g_{2}(y)}{{G_{2}}(t)}\Big{)}\bigg{)}^{2}dy-[\overline{\mathcal{H}}^{y}(X;t)]^{2}.$
(3.7)

Now, using $x=[{G_{1}}(y)]^{a}$

$\displaystyle\int_{0}^{t}\frac{g_{2}(y)}{{G_{2}}(t)}\bigg{(}y\log\Big{(}\frac{g_{2}(y)}{{G_{2}}(t)}\Big{)}\bigg{)}^{2}dy$
$\displaystyle=$
$\displaystyle\frac{1}{[{G_{1}}(t)]^{a}}\bigg{\\{}\int_{0}^{[{G_{1}}(t)]^{a}}[\mathcal{L}(x:a)]^{2}dx$
(3.8)
$\displaystyle+2\Lambda^{*(a)}(t)\int_{0}^{[{G_{1}}(t)]^{a}}{G_{1}}^{-1}(x^{1/a})\mathcal{L}(x:a)dx$
$\displaystyle+(\Lambda^{*(a)}(t))^{2}\int_{0}^{[{G_{1}}(t)]^{a}}[{G_{1}}^{-1}(x^{1/a})]^{2}dx\bigg{\\}}$
$\displaystyle=$
$\displaystyle\frac{1}{[{G_{1}}(t)]^{a}}\int_{0}^{[{G_{1}}(t)]^{a}}[\mathcal{J}(x:a,t)]^{2}dx.$

Using (3.4) and (3.8) in (3.7), the result readily follows. ∎

Next, using Theorem 3.1, we obtain the WPVE for a PRHR model.

######  Example 3.1.

Consider the power distribution with CDF
$G_{1}(y)=(\frac{y}{\beta})^{\alpha},~{}y>0,~{}\alpha>0$ and $\beta>0$. The
WPVE for PRHR model with baseline distribution as power distribution is

$\displaystyle\overline{\mathcal{VE}}^{y}(X;t)$ $\displaystyle=\frac{a\alpha
t^{2}}{\beta^{2}(2+a\alpha)}\bigg{[}\Big{\\{}\log\bigg{(}\Big{(}\frac{a\alpha\beta^{a\alpha-1}}{t^{a\alpha}}\Big{)}^{\beta}\bigg{)}\Big{\\}}^{2}+2\beta(a\alpha-1)\Big{(}\log(t/\beta)-\frac{1}{2+a\alpha}\Big{)}$
$\displaystyle\times\log\bigg{(}\Big{(}\frac{a\alpha\beta^{a\alpha-1}}{t^{a\alpha}}\Big{)}^{\beta}\bigg{)}+\frac{2\beta^{2}(a\alpha-1)^{2}}{(2+a\alpha)}\Big{\\{}(1+a\alpha)\log(t/\beta)-\frac{1}{2+a\alpha}\Big{\\}}\bigg{]}$
$\displaystyle-\Big{(}\frac{\beta}{t}\Big{)}^{2a\alpha}\Big{(}\frac{t^{1+a\alpha}}{\beta^{a\alpha(1+a\alpha)}}\Big{)}^{2}\bigg{[}a\alpha\Big{\\{}\log(a\alpha/\beta)-a\alpha\log(t/\beta)\Big{\\}}+\frac{a\alpha-1}{1+a\alpha}\bigg{]}^{2}.$
(3.9)

We have plotted the WPVE in (3.1) in Figure 5 with respect to $t$ and $a$.

Figure 5:  Plots for the WPVE $(a)$ with respect to $t$ (for fixed
$a,\alpha,\beta$) and $(b)$ with respect to $a$ (for fixed $t,\alpha,\beta$)
in Example 3.1.

##  4 Weighted paired dynamic varentropy

Let $Y$ be a discrete RV with mass function $p_{i}$, $i=1,\ldots,n$. Then, the
paired entropy is (see Burbea and Rao (1982))

$\displaystyle\mathcal{PH}=-\sum_{i=1}^{n}\Big{[}(1-p_{i})\log(1-p_{i})+p_{i}\log(p_{i})\Big{]}.$
(4.1)

Motivated by the paired entropy, Klein et al. (2016) introduced cumulative
paired entropy for a continuous RV, which is given by

$\displaystyle\mathcal{CP}=-\int_{0}^{\infty}[G(y)\log(G(y))+\bar{G}(y)\log(\bar{G}(y))]dy.$
(4.2)

Note that $\mathcal{CP}$ in (4.2) is a combination of the cumulative entropy
(see Di Crescenzo and Longobardi (2009)) and cumulative residual entropy (see
Rao et al. (2004)). Further, Klein et al. (2016) studied its properties.
Motivated by the concept of the paired entropy and cumulative paired entropy,
here, we introduce a new information measure combining the concept of past
entropy and residual entropy. Suppose $Y$ is a RV with CDF $G(\cdot)$. Then,
the WPDE for $t>0$ is defined as

$\displaystyle\mathcal{PH}^{\omega}(Y;t)$
$\displaystyle=-\left[\int_{0}^{t}\omega(y)\frac{g(y)}{G(t)}\log\left(\frac{g(y)}{G(t)}\right)dy+\int_{t}^{\infty}\omega(y)\frac{g(y)}{\bar{G}(t)}\log\left(\frac{g(y)}{\bar{G}(t)}\right)dy\right]$
$\displaystyle=\overline{\mathcal{H}}^{\omega}(Y;t)+\mathcal{H}^{\omega}(Y;t).$
(4.3)

Consider an affine transformation $X=\alpha Y+\beta$, where $\alpha>0$,
$\beta\geq 0$. Then, for $\omega(y)=y$, the WPDE is obtained as

$\mathcal{PH}^{y}(X;t)=\mathcal{PH}^{\omega_{2}}\Big{(}Y;\frac{t-\beta}{\alpha}\Big{)}+\log(\alpha)\Big{\\{}E\Big{[}\alpha
Y+\beta\Big{|}Y\leq\frac{t-\beta}{\alpha}\Big{]}+E\Big{[}\alpha
Y+\beta\Big{|}Y\geq\frac{t-\beta}{\alpha}\Big{]}\Big{\\}},$ (4.4)

where $\omega_{2}(y)=\alpha y+\beta.$ From (4.4), we observe that like
weighted dynamic (residual and past) entropies, the WPDE is also shift-
dependent. Next, the closed form expression of the WPDE is obtained.

######  Example 4.1.

  * $(i)$

For the uniform RV $Y$ with CDF $G(y)=\frac{y}{\beta},~{}y\in[0,\beta]$ and
$\beta>0$, the WPDE is

$\displaystyle\mathcal{PH}^{y}(Y;t)$
$\displaystyle=\frac{t}{2}\log(t)+\frac{\beta+t}{2}\big{(}\log(\beta-t)\big{)},~{}t>0.$
(4.5)

  * $(ii)$

Assume that $Y$ follows exponential distribution with mean $1/\lambda$. For
$t>0,$

$\displaystyle\mathcal{PH}^{y}(Y;t)$
$\displaystyle=\frac{1}{\lambda(e^{-\lambda
t}-1)}\Big{[}\Big{\\{}1-e^{-\lambda t}(1+\lambda
t)\Big{\\}}\Big{\\{}\log\Big{(}\frac{\lambda}{1-e^{-\lambda
t}}\Big{)}-2\Big{\\}}+\lambda^{2}t^{2}e^{-\lambda(t)}\Big{]}$
$\displaystyle+\frac{1}{\lambda}\Big{\\{}(\lambda t+1)\log(\lambda)-\lambda
t-2\Big{\\}}.$ (4.6)

To see the behaviour of the WPDE for uniform and exponential distributions, we
have plotted their WPDE in Figure 6. The graphs show that the WPDE is non-
monotone for these distributions.

Figure 6:  Plots of the WPDE for $(a)$ uniform distribution (for fixed $b$)
and $(b)$ exponential distribution (for fixed $\lambda$) as in Example
4.1$(i)$ and Example 4.1$(ii)$, respectively.

Inspired by the notions of the paired, cumulative paired, and weighted paired
dynamic entropies, here we propose the concept of the WPDVE.

######  Definition 4.1.

Suppose $Y$ is a RV with CDF $G(\cdot)$, PDF $g(\cdot),$ and survival function
$\bar{G}(\cdot)$. The WPDVE of $Y$ is defined as

$\displaystyle\mathcal{PVE}^{\omega}(Y;t)=\overline{\mathcal{VE}}^{\omega}(Y;t)+\mathcal{VE}^{\omega}(Y;t),$
(4.7)

where $\overline{\mathcal{VE}}^{\omega}(Y;t)$ and $\mathcal{VE}^{\omega}(Y;t)$
are respectively known as the WPVE (see (2.1)) and WRVE (see Saha and Kayal
(2024)).

Note that

$\displaystyle\overline{\mathcal{VE}}^{\omega}(Y;t)+\mathcal{VE}^{\omega}(Y;t)$
$\displaystyle=$ $\displaystyle
Var\Big{(}-\omega(Y)\log(g^{*}_{t}(Y))\Big{)}+Var\Big{(}-\omega(Y)\log(g_{t}(Y))\Big{)}$
(4.8) $\displaystyle=$ $\displaystyle
Var\Big{(}-\omega(Y)\log(g^{*}_{t}(Y)\times g_{t}(Y))\Big{)}$ $\displaystyle=$
$\displaystyle Var\Big{(}IC_{J}^{\omega}(Y)\Big{)},$

where $IC_{J}^{\omega}(y)=-\omega(y)\log(g^{*}_{t}(y)\times g_{t}(y))$ is the
combined IC of past and residual random lifetimes. In the following, we
express WPDVE in terms of the conditional expectations, WPDE, and weighted
dynamic (residual and past) entropies when $\omega(y)=y$:

$\displaystyle\mathcal{PVE}^{y}(Y;t)=$ $\displaystyle
E[(Y\log(g(Y))^{2})|Y<t]+E[(Y\log(g(Y))^{2})|Y>t]-[{\mathcal{PH}}^{y}(Y;t)]^{2}$
$\displaystyle-2\Lambda(t){\mathcal{H}}^{y^{2}}(Y;t)-2\Lambda^{*}(t)\overline{\mathcal{H}}^{y^{2}}(Y;t)-(\Lambda^{*}(t))^{2}E[Y^{2}|Y\leq
t]$
$\displaystyle-(\Lambda(t))^{2}E[Y^{2}|Y>t]+2{\mathcal{H}}^{y}(Y;t)\overline{{\mathcal{H}}}^{y}(Y;t).$
(4.9)

When $t\rightarrow 0$ or $t\rightarrow\infty,$ the WPDVE in (4.7) reduces to
the weighted varentropy, which has been studied by Saha and Kayal (2024).
Further, considering $\omega(y)=1$, the WPDVE becomes usual varentropy (see
Fradelizi et al. (2016)) when $t\rightarrow 0$ or $t\rightarrow\infty.$ Due to
these reasons, the newly proposed information measure in Definition 4.1 can be
treated as a generalised information measure. Next, we establish a lower bound
of the WPDVE via the WPVE and WRVE.

######  Theorem 4.1.

Suppose $Y$ is a RV. Then, for a general weight function $\omega(.)$, we have

$\displaystyle\mathcal{PVE}^{\omega}(Y;t)\geq\max\\{\overline{\mathcal{VE}}^{\omega}(Y;t),\mathcal{VE}^{\omega}(Y;t)\\},~{}t>0.$
(4.10)

###### Proof.

From (4.7), it is clear that

$\displaystyle\mathcal{PVE}^{\omega}(Y;t)\geq\overline{\mathcal{VE}}^{\omega}(Y;t)>0$
(4.11)

and

$\displaystyle\mathcal{PVE}^{\omega}(Y;t)\geq\mathcal{VE}^{\omega}(Y;t)>0.$
(4.12)

Now, combining (4.11) and (4.12), the result follows. ∎

The following theorem provides an upper bound of the WPDVE.

######  Theorem 4.2.

For the RV $Y$

$\displaystyle\mathcal{PVE}^{y}(Y;t)\leq$ $\displaystyle
E[(\psi_{1}(Y))^{2}|Y\leq t]+E[(\psi_{1}(Y))^{2}|Y\geq
t]-2\Lambda^{*}(t)\overline{\mathcal{H}}^{y^{2}}(Y;t)-2\Lambda(t){\mathcal{H}}^{y^{2}}(Y;t),$

where $t>0$ and $\psi_{1}(y)=y\log(g(y))$.

###### Proof.

From (2.2), we obtain

$\displaystyle\mathcal{\overline{VE}}^{y}(Y;t)\leq E[(\psi_{1}(Y))^{2}|Y\leq
t]-2\Lambda^{*}(t)\overline{\mathcal{H}}^{y^{2}}(Y;t).$ (4.13)

Further, from $(3.2)$ of Saha and Kayal (2024), we get an upper bound of the
WRVE as

$\displaystyle\mathcal{{VE}}^{y}(Y;t)\leq E[(\psi_{1}(Y))^{2}|Y\geq
t]-2\Lambda(t)\overline{\mathcal{H}}^{y^{2}}(Y;t).$ (4.14)

Thus, the required bound follows after summing (4.13) and (4.14). ∎

######  Theorem 4.3.

Suppose $Y^{*}_{t}$ and $Y_{t}$ respectively denote the past and residual
lifetimes with finite MRL $\mu(t)$, finite MPL $\mathcal{M}(t)$, finite VRL
$\sigma^{2}_{1}(t)$, and VPL $\sigma^{2}(t)$. Then,

$\displaystyle\mathcal{PVE}^{y}(Y;t)\geq\max\\{\pi(y,t),\theta(y,t)\\}$ (4.15)

where
$\pi(y,t)=\sigma^{2}(t)\\{1+E[-\zeta_{t}(Y_{t})\log\big{(}g_{t}(Y_{t}\big{)}]+E[Y_{t}\zeta^{{}^{\prime}}_{t}(Y_{t})]\\}^{2}$
and
$\theta(y,t)=\sigma^{2}_{1}(t)\\{1+E[-\eta_{t}(Y_{t})\log\big{(}g_{t}(Y_{t}\big{)}+E[Y_{t}\eta^{{}^{\prime}}_{t}(Y_{t})]\\}^{2}$.
Here, $\eta_{t}(y)$ and $\zeta_{t}(y)$ are real-valued functions, respectively
obtained from

$\sigma^{2}_{1}(t)\eta_{t}(y)g_{t}(y)=\int_{0}^{y}(\mu(t)-u)g_{t}(u)du,~{}y>0$

and

$\displaystyle\sigma^{2}(t)\zeta_{t}(y)g^{*}_{t}(y)=\int_{0}^{y}(\mathcal{M}(t)-u)g^{*}_{t}(u)du,~{}y>0.$

###### Proof.

From Theorem 2.2, we have

$\displaystyle\overline{\mathcal{VE}}^{y}(Y;t)\geq\sigma^{2}(t)\\{1+E[-\zeta_{t}(Y_{t})\log(g^{*}_{t}(Y_{t}))]+E[Y_{t}\zeta^{{}^{\prime}}_{t}(Y_{t})]\\}^{2}.$
(4.16)

Further, from Theorem $3.3$ of Saha and Kayal (2024), we get

$\displaystyle\mathcal{VE}^{y}(Y;t)\geq\sigma^{2}_{1}(t)\\{1+E[-\eta_{t}(Y_{t})\log(g_{t}(Y_{t}))]+E[Y_{t}\eta^{{}^{\prime}}_{t}(Y_{t})]\\}^{2}.$
(4.17)

Thus, using (4.16) and (4.17), the result readily follows. ∎

This section ends with a result dealing with the effect of the WPDVE under
affine transformations.

######  Theorem 4.4.

Let $Y$ be a RV. Assume that $X=aY+b$ with $a>0,~{}b\geq 0$. Then, for all
real number $t>0$,

$\displaystyle\mathcal{PVE}^{y}(X;t)$
$\displaystyle=\mathcal{PVE}^{\omega_{1}}(X;t)-2\log(a)\mathcal{PH}^{\omega_{1}}(X;t)+(\log(a))^{2}[\xi(a,b,t)\\{1-\xi(a,b,t)\\}$
$\displaystyle+\varphi(a,b,t)\\{1-\varphi(a,b,t)\\}]-2\log(a)[\mathcal{H}^{\omega_{1}}(Y;(t-b)/a)\\{1+\varphi(a,b,t)\\}$
$\displaystyle+\overline{\mathcal{H}}^{\omega_{1}}(Y;(t-b)/a)\\{1+\xi(a,b,t)\\}],$

where $\mathcal{PVE}^{\omega_{1}}(X;t)$ and $\mathcal{PH}^{\omega_{1}}(X;t)$
are the WPDVE and WPDE with weight function
$\omega_{1}\equiv\omega_{1}(y)=ay+b$ respectively, and
$\varphi(a,b,t)=E[aY+b|Y>(t-b)/a]$ and $\xi(a,b,t)=E[aY+b|Y\leq(t-b)/a]$.

###### Proof.

From Corollary 2.2 and Corollary $3.1$ of Saha and Kayal (2024), the proof
follows. ∎

##  5 Estimation of the WPVE and WPDVE

This section presents kernel-based non-parametric estimates of the WPVE and
WPDVE. We see the performance of the proposed estimates using a Monte-Carlo
simulation study. For both WPVE and WPDVE, we have also considered parametric
estimation assuming that the data are taken from exponential population. A
data set representing average daily wind speeds is considered and analysed for
the purpose of estimating WPVE.

###  5.1 WPVE

Here, we consider non-parametric and parametric estimations of the WPVE.
First, we discuss about non-parametric estimation.

####  5.1.1 Non-parametric estimation

We introduce a non-parametric estimates based on the kernel estimates of WPVE
in (2.1). The kernel estimate of the PDF ${g}(\cdot)$ is given by

$\displaystyle\widehat{g}(y)=\frac{1}{nb_{n}}\sum_{i=1}^{n}\mathcal{K}\left(\frac{y-Y_{i}}{b_{n}}\right),$
(5.1)

where $\mathcal{K}(\cdot)$ is known as kernel, satisfying the following
properties.

  * •

It is non-negative;

  * •

$\int\mathcal{K}(y)dy=1$;

  * •

The kernel is symmetric at the origin;

  * •

It satisfies the Lipschitz condition.

In (5.1), the sequence of positive real numbers $\\{b_{n}\\}$ is known as the
bandwidths such that $b_{n}\rightarrow 0$ and $nb_{n}\rightarrow\infty,$ for
$n\rightarrow\infty$. For details about the kernel density estimates, readers
can refer to Rosenblatt (1956) and Parzen (1962). Using (5.1), a non-
parametric kernel estimate of $\overline{\mathcal{VE}}^{y}(Y;t)$ is

$\displaystyle\widehat{\overline{\mathcal{VE}}^{y}}(Y;t)=\int_{0}^{t}\widehat{\eta}(y)(y\log\widehat{\eta}(y))^{2}dy-\left[\int_{0}^{t}y\widehat{\eta}(y)\log\widehat{\eta}(y)dy\right]^{2},~{}t>0,$
(5.2)

where $\widehat{\eta}(y)=\frac{\widehat{g}(y)}{\widehat{G}(t)}$ and
$\widehat{G}(t)=\int_{0}^{t}\widehat{g}(y)dy$. Below, we conduct Monte-Carlo
simulation to see the performance of the estimate given in (5.2).

### Simulation study

A Monte-Carlo simulation study has been performed to generate data sets from
exponential distribution with mean $1/\lambda$ for different sample sizes. The
true parameter value is taken as $\lambda=0.7$. The software “Mathematica” has
been used for the simulation study. For computing the AB and MSE of the
kernel-based non-parametric estimate, we use $100$ replications. Here,
Gaussian kernel is used for estimation. It is given by

$\displaystyle\mathcal{K}(y)=\frac{1}{\sqrt{2\pi}}e^{-\frac{y^{2}}{2}}.$ (5.3)

The AB and MSE of the kernel-based non-parametric estimate
$\widehat{\overline{\mathcal{VE}}^{y}}(Y;t)$ in (5.2) have been computed and
presented in Table LABEL:tb1 for different choices of $t$ and $n$. We have
considered $t=0.1,0.2,0.3,0.4,1$ and $n=100,120,150,200.$ From Table
LABEL:tb1, we observe that in general the AB and MSE decrease as $n$
increases. This confirms the consistency of the proposed estimate
$\widehat{\overline{\mathcal{VE}}^{y}}(Y;t)$ in (5.2).

Table 1: The AB, MSE and $\overline{\mathcal{VE}}^{y}(Y;t)$ for the kernel-
based estimate of WPVE in (5.2).

$t$ | $n$ | AB | MSE | $\overline{\mathcal{VE}}^{y}(Y;t)$
---|---|---|---|---
| 100 | $0.002899$ | $0.000065$ |
120 | $0.002375$ | $0.000008$ |
0.1 | 150 | $0.002066$ | $0.000006$ | $0.004285$
| 200 | $0.001970$ | $0.000005$ |
| 100 | $0.004743$ | $0.000026$ |
120 | $0.004657$ | $0.000025$ |
0.2 | 150 | $0.004287$ | $0.000022$ | $0.007901$
| 200 | $0.004317$ | $0.000020$ |
| 100 | $0.006270$ | $0.000043$ |
120 | $0.005539$ | $0.000036$ |
0.3 | 150 | $0.005343$ | $0.000032$ | $0.009078$
| 200 | $0.005303$ | $0.000030$ |
| 100 | $0.005609$ | $0.000035$ |
120 | $0.005426$ | $0.000034$ |
0.4 | 150 | $0.005468$ | $0.000033$ | $0.008114$
| 200 | $0.005173$ | $0.000029$ |
| 100 | $~{}~{}0.002632$ | $0.000042$ |
120 | $~{}~{}0.001417$ | $0.000026$ |
1.0 | 150 | $0.000612$ | $0.000019$ | $0.011937$
| 200 | $0.000387$ | $0.000016$ |

### Real data set

Here, we consider a real data set representing average daily wind speeds (in
meter/second) in November, $2007$ at Elanora Heights, a northeastern suburb of
Sydney, Australia. The real data set is presented in Table LABEL:tb2 (see Best
et al. (2010)).

Table 2: The data set.

0.5833 0.6667 0.6944 0.7222 0.7500 0.7778 0.8056 0.8056 0.8611
---
0.8889 0.9167 1.0000 1.0278 1.0278 1.1111 1.1111 1.1111 1.1667
1.1667 1.1944 1.2778 1.2778 1.3056 1.3333 1.3333 1.3611 1.4444
2.1111 2.1389 2.7778

For checking the best fitted model for the real data set, goodness of fit test
has been applied. Here, we have considered four statistical models: Gumbel-II
(GMB-II), Weibull, generalised X-exponential (GXE), and exponential (EXP)
distributions. We use negative log-likelihood criterion ($-\ln L$), Akaikes-
information criterion (AIC), AICc, and the Bayesian information criterion
(BIC). From Table $3$, we observe that the GMB-II distribution fits better
than other distributions as the values of the test statistics are smaller than
that of the other distributions. The values of the MLEs of the unknown
parameters of GMB-II are $\alpha=3.3869$ and $\lambda=0.7544$. The Gaussian
kernel in (5.3) is employed as the kernel function for estimation purpose. The
values of AB and MSE of the proposed estimate in (5.2) have been computed
using $500$ bootstrap samples with size $n=30$ and $b_{n}=0.35$. These are
given in Table LABEL:tb4 for different choices of $t$.

Table 3: The MLEs, BIC, AICc, AIC, and negative log-likelihood values of the
statistical models for the data set presented in Table LABEL:tb2.

Model | Shape | Scale | -ln L | AIC | AICc | BIC
---|---|---|---|---|---|---
GMB-II | $\widehat{\alpha}=3.3869$ | $\widehat{\lambda}=0.7544$ | 12.5333 | 29.0665 | 29.5109 | 31.8689
Weibull | $\widehat{\alpha}=2.5393$ | $\widehat{\lambda}=1.3048$ | 18.5659 | 41.1317 | 41.5762 | 43.9341
GXE | $\widehat{\alpha}=4.1464$ | $\widehat{\lambda}=1.1421$ | 17.3245 | 38.6491 | 39.0935 | 41.4515
EXP | $\widehat{\lambda}=0.8633$ |  | 34.4095 | 70.8191 | 70.9620 | 72.2203

Table 4: The AB, MSE and ${\it VE}^{y}(Y;t)$ for the data set in Table $2$.

$t$ |  AB |  MSE |  $\mathcal{\overline{VE}}^{y}(Y;t)$
---|---|---|---
1.0 |  0.05122 |  0.00453 |  0.12205
1.1 |  0.02278 |  0.00135 |  0.08995
1.2 |  0.03233 |  0.00176 |  0.07566
1.3 |  0.05258 |  0.00314 |  0.08479
1.4 |  0.09765 |  0.00959 |  0.12025
1.5 |  0.14900 |  0.02237 |  0.18279
1.8 |  0.33773 |  0.11743 |  0.52167
2.0 |  0.45306 |  0.21668 |  0.85184
2.5 |  0.59685 |  0.37733 |  1.98038
3.0 |  0.65917 |  0.55132 |  3.10000

####  5.1.2 Parametric estimation

Here, we consider parametric estimation of the WPVE. We assume that the data
are taken from an exponential population with parameter $\lambda.$ In this
case, the WPVE is obtained as

$\displaystyle\mathcal{\overline{VE}}^{y}(Y;t)=\int_{0}^{t}y^{2}\frac{\lambda
e^{-\lambda y}}{1-e^{-\lambda t}}\bigg{(}\log\Big{(}\frac{\lambda e^{-\lambda
y}}{1-e^{-\lambda t}}\Big{)}\bigg{)}^{2}dy-\int_{0}^{t}y\frac{\lambda
e^{-\lambda y}}{1-e^{-\lambda t}}\log\Big{(}\frac{\lambda e^{-\lambda
y}}{1-e^{-\lambda t}}\Big{)}dy.$ (5.4)

We apply maximum likelihood estimation technique for the purpose of estimation
of (5.4). Let $\hat{\lambda}$ be the maximum likelihood estimate (MLE) of the
model parameter $\lambda$. Using the invariance property, the MLE of the WPVE
is obtained as

$\displaystyle\mathcal{\widetilde{\overline{VE}}}^{y}(Y;t)=\int_{0}^{t}y^{2}\frac{\hat{\lambda}e^{-\hat{\lambda}y}}{1-e^{-\hat{\lambda}t}}\bigg{(}\log\Big{(}\frac{\hat{\lambda}e^{-\hat{\lambda}y}}{1-e^{-\hat{\lambda}t}}\Big{)}\bigg{)}^{2}dy-\int_{0}^{t}y\frac{\hat{\lambda}e^{-\hat{\lambda}y}}{1-e^{-\hat{\lambda}t}}\log\Big{(}\frac{\hat{\lambda}e^{-\hat{\lambda}y}}{1-e^{-\hat{\lambda}t}}\Big{)}dy,~{}t>0.$
(5.5)

Here, we conduct Monte-Carlo simulation using $R$ software to see the
behaviour of the proposed parametric estimate of WPVE. We take $\lambda=0.7$
as its true parameter value. We have considered $t=0.1,0.2,0.3,0.4,1$ and
$n=100,120,150,200.$ Using $100$ replications, the AB and MSE have been
computed and presented in Table LABEL:tb5. From Table LABEL:tb5, we observe
that MSEs decrease as $n$ increases.

From Table LABEL:tb1 and Table LABEL:tb5, we observe that the performance of
the estimate of the WPVE using parametric approach is superior than the non-
parametric approach in terms of the AB and MSE values.

Table 5: The AB, MSE, and $\overline{\mathcal{VE}}^{y}(Y;t)$ for the
parametric estimator of WPVE in (5.5).

$t$ | $n$ | AB | MSE | $\overline{\mathcal{VE}}^{y}(Y;t)$
---|---|---|---|---
| 100 | $1.977\times 10^{-6}$ | $1.653\times 10^{-10}$ |
120 | $1.961\times 10^{-6}$ | $1.240\times 10^{-10}$ |
0.1 | 150 | $2.387\times 10^{-6}$ | $1.023\times 10^{-10}$ | $0.004285$
| 200 | $1.181\times 10^{-6}$ | $6.874\times 10^{-11}$ |
| 100 | $1.049\times 10^{-5}$ | $4.766\times 10^{-9}$ |
120 | $1.044\times 10^{-5}$ | $3.578\times 10^{-9}$ |
0.2 | 150 | $1.275\times 10^{-5}$ | $2.953\times 10^{-9}$ | $0.007901$
| 200 | $6.293\times 10^{-6}$ | $1.987\times 10^{-9}$ |
| 100 | $2.421\times 10^{-5}$ | $2.647\times 10^{-8}$ |
120 | $2.424\times 10^{-5}$ | $1.989\times 10^{-8}$ |
0.3 | 150 | $2.977\times 10^{-5}$ | $1.641\times 10^{-8}$ | $0.009078$
| 200 | $1.463\times 10^{-5}$ | $1.108\times 10^{-8}$ |
| 100 | $3.765\times 10^{-5}$ | $6.900\times 10^{-8}$ |
120 | $3.809\times 10^{-5}$ | $5.193\times 10^{-8}$ |
0.4 | 150 | $4.728\times 10^{-5}$ | $4.284\times 10^{-8}$ | $0.008114$
| 200 | $2.305\times 10^{-5}$ | $2.908\times 10^{-8}$ |
| 100 | $4.220\times 10^{-4}$ | $5.061\times 10^{-6}$ |
120 | $3.981\times 10^{-4}$ | $3.758\times 10^{-6}$ |
1.0 | 150 | $4.595\times 10^{-4}$ | $3.105\times 10^{-6}$ | $0.008114$
| 200 | $2.369\times 10^{-4}$ | $2.015\times 10^{-6}$ |

###  5.2 WPDVE

In this subsection, we have proposed non-parametric and parametric estimates
of the WPDVE. Below, we discuss the non-parametric estimate.

####  5.2.1 Non-parametric estimation

Similar to (5.2), a kernel-based non-parametric estimate of the WPDVE in (4.7)
is obtained as

$\displaystyle\widehat{\mathcal{PVE}}^{y}(Y;t)$
$\displaystyle=\int_{0}^{t}\widehat{\eta}(y)(y\log(\widehat{\eta}(y)))^{2}dy+\int_{t}^{\infty}\widehat{\delta}(y)(y\log(\widehat{\delta}(y)))^{2}dy-\left[\int_{0}^{t}y\widehat{\eta}(y)\log(\widehat{\eta}(y))dy\right]^{2}$
$\displaystyle~{}~{}-\left[\int_{t}^{\infty}y\widehat{\delta}(y)\log(\widehat{\delta}(y))dy\right]^{2},$
(5.6)

where $\widehat{\delta}()=\frac{\widehat{g}(y)}{\widehat{\bar{G}}(t)}$,
$\widehat{\eta}(y)=\frac{\widehat{g}(y)}{\widehat{G}(t)}$,
$\widehat{\bar{G}}(t)=\int_{t}^{\infty}\widehat{g}(y)dy$, and
$\widehat{G}(t)=\int_{0}^{t}\widehat{g}(y)dy$.

### Simulation study

Similar to the preceding subsection, here a Monte-Carlo simulation study has
been carried out to check the performance of the proposed kernel-based non-
parametric estimate of the WPDVE given in (5.2.1). The data set has been
generated from exponential distribution with $\lambda=5$ using “Mathematica”
software. For different values of $n=100,120,150,200$ and
$t=0.05,0.1,0.15,0.2$, the AB and MSE values have been computed using $500$
replications. We have used the Gaussian kernel given in (5.3). The computed
values of the AB and MSE are presented in Table LABEL:tb6. From Table
LABEL:tb6, we notice similar observation to the case of WPVE.

Table 6: The AB, MSE, and $\mathcal{PVE}^{y}(Y;t)$ for the kernel estimator of
WPDVE in (5.2.1).

$t$ | $n$ | AB | MSE | $\mathcal{PVE}^{y}(Y;t)$
---|---|---|---|---
| 100 | $0.15322$ | $0.04596$ |
120 | $0.15714$ | $0.04499$ |
0.05 | 150 | $0.13113$ | $0.03731$ | $0.44062$
| 200 | $0.11894$ | $0.03376$ |
| 100 | $0.19768$ | $0.06524$ |
120 | $0.187725$ | $0.06388$ |
0.10 | 150 | $0.16648$ | $0.05171$ | $0.49772$
| 200 | $0.14099$ | $0.04369$ |
| 100 | $0.23206$ | $0.08678$ |
120 | $0.20336$ | $0.07177$ |
0.15 | 150 | $0.20001$ | $0.06897$ | $0.55890$
| 200 | $0.18718$ | $0.06399$ |
| 100 | $0.26717$ | $0.11190$ |
120 | $0.26187$ | $0.10100$ |
0.20 | 150 | $0.23457$ | $0.09309$ | $0.62414$
| 200 | $0.22166$ | $0.07981$ |

####  5.2.2 Parametric estimation

Consider an exponential population with mean $1/\lambda,$ $\lambda>0.$ The
WPDVE of the exponential distribution is

$\displaystyle\mathcal{PVE}^{y}(Y;t)$
$\displaystyle=\int_{0}^{t}y^{2}\frac{\lambda e^{-\lambda y}}{1-e^{-\lambda
t}}\bigg{(}\log\Big{(}\frac{\lambda e^{-\lambda y}}{1-e^{-\lambda
t}}\Big{)}\bigg{)}^{2}dy-\int_{0}^{t}y\frac{\lambda e^{-\lambda
y}}{1-e^{-\lambda t}}\log\Big{(}\frac{\lambda e^{-\lambda y}}{1-e^{-\lambda
t}}\Big{)}dy$ $\displaystyle+\int_{t}^{\infty}\lambda
e^{(t-y)\lambda}\Big{(}y\log(\lambda
e^{(t-y)\lambda})\Big{)}^{2}dy-\int_{t}^{\infty}\lambda
ye^{(t-y)\lambda}\log(\lambda e^{(t-y)\lambda})dy.$ (5.7)

Firstly, we estimate the model parameter $\lambda$ using maximum likelihood
estimation technique for estimating (5.2.2). The MLE of
$\mathcal{PVE}^{y}(Y;t)$ in (5.2.2) is

$\displaystyle\widetilde{\mathcal{PVE}}^{y}(Y;t)=$
$\displaystyle=\int_{0}^{t}y^{2}\frac{\hat{\lambda}e^{-\hat{\lambda}y}}{1-e^{-\hat{\lambda}t}}\bigg{(}\log\Big{(}\frac{\hat{\lambda}e^{-\hat{\lambda}y}}{1-e^{-\hat{\lambda}t}}\Big{)}\bigg{)}^{2}dy-\int_{0}^{t}y\frac{\hat{\lambda}e^{-\hat{\lambda}y}}{1-e^{-\hat{\lambda}t}}\log\Big{(}\frac{\hat{\lambda}e^{-\hat{\lambda}y}}{1-e^{-\hat{\lambda}t}}\Big{)}dy$
$\displaystyle+\int_{t}^{\infty}\hat{\lambda}e^{(t-y)\hat{\lambda}}\Big{(}y\log(\hat{\lambda}e^{(t-y)\hat{\lambda}})\Big{)}^{2}dy-\int_{t}^{\infty}\hat{\lambda}ye^{(t-y)\hat{\lambda}}\log(\hat{\lambda}e^{(t-y)\hat{\lambda}})dy,$
(5.8)

where $\hat{\lambda}$ is the MLE of $\lambda$. To evaluate the performance of
the proposed parametric estimate, Monte-Carlo simulation is conducted using
$R$ software with $500$ replications. Here, we consider the true value of
$\lambda$ as $5$. For sample sizes $n=100,120,150$ and $200$, the AB and MSE
values have been presented in Table LABEL:tb7 for different choices of $t$ .
From Table LABEL:tb7, we observe that MSEs decrease as $n$ increases, which
assures the consistency and validation of the propose estimate
$\widetilde{\mathcal{PVE}}^{y}(Y;t)$ in (5.2.2).

From Table LABEL:tb6 and Table LABEL:tb7, we observe that the parametric
estimate in (5.2.2) performs better than the non-parametric estimate in
(5.2.1) when the data are generated from exponential distribution with
$\lambda=5$ in terms of the AB and MSE.

Table 7: The AB, MSE, and $\mathcal{PVE}^{y}(Y;t)$ for parametric estimate of
WPDVE in (5.2.2).

$t$ | $n$ | AB | MSE | $\mathcal{PVE}^{y}(Y;t)$
---|---|---|---|---
| 100 | $0.00884$ | $0.01091$ |
120 | $0.00923$ | $0.00927$ |
0.05 | 150 | $0.00814$ | $0.00712$ | $0.44062$
| 200 | $0.00618$ | $0.00526$ |
| 100 | $0.00905$ | $0.01236$ |
120 | $0.00953$ | $0.01049$ |
0.10 | 150 | $0.00843$ | $0.00807$ | $0.49772$
| 200 | $0.00641$ | $0.00597$ |
| 100 | $0.00927$ | $0.01390$ |
120 | $0.00982$ | $0.01181$ |
0.15 | 150 | $0.00872$ | $0.00909$ | $0.55890$
| 200 | $0.00663$ | $0.00672$ |
| 100 | $0.00948$ | $0.01554$ |
120 | $0.01012$ | $0.01319$ |
0.20 | 150 | $0.00901$ | $0.01017$ | $0.62414$
| 200 | $0.00686$ | $0.00752$ |

##  6 Conclusions

In this work, we have introduced WPVE and discussed its various properties.
Bounds of the WPVE have been obtained. Due to difficult structure of a
distribution, sometimes it is very tough to obtain explicit expression of WPVE
of a transformed RV. Here, we have obtained a result which will be useful in
obtaining the WPVE of a transformed RV under monotone transformations. We have
also introduced WPVE for PRHR model and explore some properties. Several
examples have been considered for the purpose of illustration. Further, we
proposed WPDE and WPDVE, and studied their several properties. It is observed
that the WPDVE is a generalisation of the weighted varentropy and varentropy.
The effectiveness of the WPDVE under affine transformations has been
investigated. Lower and upper bounds of the WPDVE are derived. Furthermore,
kernel-based non-parametric estimates for the WPVE and WPDVE have been
proposed. A simulation study is caried out to see the performance of the
proposed non-parametric estimates. In order to compare the non-parametric
estimation method with the parametric estimation method, we have considered
parametric estimation of both WPDE and WPDVE. It is noticed that the
parametric estimation method provides a better result than the non-parametric
estimation method in the terms of the AB and MSE values when the data are
generated from exponential distribution. Finally, a real data set representing
the average wind speed has been considered and analysed for the purpose of
estimation of the WPVE.

## Acknowledgements

The author Shital Saha thanks the UGC (Award No. $191620139416$), India, for
financial assistantship received to carry out this research work.

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