1. Introduction
The concept of extropy and its use has been explored rapidly in the recent years. It measures the uncertainty contained in the probability distributions and is considered as the complimentary dual of entropy introduced in [
1]. The entropy measure is shift-independent, that is, it is the same for both
X and
and it cannot be applied in some fields such as neurology. Thus, in [
2], the notion of weighted entropy measure was introduced. The authors pointed out that occurrence of an event has an impact on uncertainty in two ways. It presents both quantitative and qualitative information. That is, it initially reveals the probability of an event occurring and later demonstrates its efficacy in achieving qualitative features of a goal. It is important to note that the information obtained when a device fails to operate or a neuron fails to release spikes in a specific time interval differs significantly from the information obtained when such events occur in other equally wide intervals. This is why there is a need, in some cases, to employ a shift-dependent information measure that assigns varying measures to these distributions.The importance of the presence of weighted measures of uncertainty was exhibited in [
3].
The concept of extropy for a continuous
X has been presented and discussed across numerous works in the literature. The differential extropy defined by [
4] is
One can refer to [
5] for the extropy properties of order statistics and record values. The applications of extropy in automatic speech recognition can be found in [
6]. Various literature sources have presented a range of extropy measures and their extensions. Analogous to weighted entropy, in [
7], the concept of weighted extropy was introduced (
) in the literature. It is given as
Variable
x in the integral emphasizes the weight related to the occurrence of event
. Here, it assigns more significance to large values of
X. In the literature, extropy, its different versions and their applications have been studied by several authors (see, for instance, [
8,
9,
10]). In particular, a unified version of extropy in classical theory and in Dempster–Shafer theory was studied in [
11].
There are several papers available in the literature that delve into the estimation of extropy and its various versions. Kernel estimation on the functionals of the density function was proposed in [
12]. The optimal bandwidth for kernel density functionals is provided in [
13]. In [
14], a brief explanation was established on optimal bandwidth estimators of kernel density functionals for contaminated data. In [
15], estimators of extropy were proposed, and also its application was worked on by testing uniformity. In [
16], the concept of length biased sampling in estimating extropy was approached. Research on non-parametric estimation using dependent data is also well-explored in the literature. Work by [
17] explained the recursive and non-recursive kernel estimation of negative cumulative extropy under the
-mixing dependence condition. Recently, in [
18], the kernel estimation of the extropy function was discussed using
-mixing-dependent data. Moreover, in [
19], the log kernel estimation of extropy was introduced.
Even if there are several works available in the literature related to the estimation of extropy, little has been published on
and its estimation until now. There are situations in which we are forced to use
instead of extropy. Unlike extropy, the qualitative characteristics of information are also represented here. In [
20], the significance of employing
as opposed to regular extropy in certain scenarios was demonstrated. There are instances where certain distributions possess identical extropy values but exhibit distinct
values. In such situations, it becomes necessary to opt for
. The estimators of
can also be used in the selection of models in the reliability analysis. Here, we tried to find some estimators for
and validated it using simulation study and data analysis.
The paper is organized as follows: In
Section 2, we introduce the log kernel estimation of
. In
Section 3, an empirical kernel smoothed estimator of
is given. A simulation study is conducted to evaluate the estimators, and we also compare log kernel to kernel estimators of
in
Section 4.
Section 5 is devoted to the real data analysis to examine the proposed estimators. Finally, we conclude the study in
Section 6.
2. Log Kernel Estimation of Weighted Extropy
In this section, we introduce the concept of log kernel-based estimation of .
Let us define an
X with unknown
. We assume that
X is defined on
R and
is continuously differentiable. We suppose
is a sequence of identically distributed
. The most commonly used estimator of
is the kernel density estimator (
), given by [
21,
22] as
where
is the kernel function which satisfies the following conditions:
Here, bandwidth parameter → 0 and → as n → .
When probability density functions are estimated in a non-parametric way, standard
is frequently used. However, when we deal with data that fit distributions with heavy tails, multiple modes, or skewness, particularly those with positive values these estimators may lose their effectiveness. In all of these scenarios, applying a transformation, we can yield more consistent results. Such transformation involves a logarithmic transformation to create a non-parametric
. An important aspect of the logarithmic transformation is its ability to compress the right tail of the distribution. The obtained
are called logarithmic
(denoted as
) (refer to [
23]). Let us define
,
;
and let
be the
of
Y. The
is defined as
where
is the log kernel function with bandwidth
at location parameter
z. For any
,
satisfies conditions
for all
and
.
For any
,
where
.
We let
be a sample of identically distributed observations. We obtain the
for
by using the estimator defined in Equation (
4).
The
for the
function is
which again can be alternatively expressed as
The following theorem gives the expression for bias and variance of the
of
.
Theorem 1. Assume that the conditions given in Section 2 are satisfied in the case of log kernel function and bandwidth . Then, the bias and variance of are given, respectively, aswhere . Proof. The proof is omitted as it is similar to [
19]. □
The following theorem shows that the proposed estimator is consistent.
Theorem 2. is a consistent estimator of , where and are defined in Equations (2) and (7). Also, let be the log kernel function and be the bandwidth which satisfies the conditions given in Section 2. Then, we can say that, as n tends to Proof. Since the proof is similar to that of [
19], it is omitted. □
The below theorem shows that the of is integratedly uniformly consistent in the quadratic mean estimator of .
Theorem 3. Consider log kernel function and bandwidth parameter that fulfills the conditions outlined in Section 2. If is according to Equation (7), then is integratedly uniformly consistent in the quadratic mean estimator of . Proof. As the proof resembles that of [
19], it is omitted here. □
Here, we provide the expression for the optimal bandwidth of .
Optimal Bandwidth
Here, we offer the expression for the optimal bandwidth using mean integrated square error (
). The
of
is given as
Using the expression for bias and variance given in Equations (
9) and (
10), the
of
is given as
The asymptotic
(
) can be obtained by ignoring the higher-order terms and is given as
The optimal bandwidth is then attained after minimizing
with respect to
, and it is given by
4. Simulation Study
We manage a simulation study to evaluate the performance of the presented estimators. Random samples are generated corresponding to different sample sizes from some standard distributions, and then both bias and root mean square (
) are calculated for 10,000 samples. Bandwidth parameter
is determined using the plug-in method as proposed in [
26].
To enable a comparison between
and
of
, we again propose a
for
using Equation (
3). The estimator is given by
where
is the
given in [
21]. Using the consistency property of the
, it is clear that the proposed estimator in Equation (
18) for
is also consistent. To lay the ground work for comparison, we generate samples from exponential distribution, log normal distribution, a heavy-tailed distribution and uniform distribution. The Gaussian log transformed kernel and the Gaussian kernel are the kernel functions used for simulation.
From the above
Table 3,
Table 4 and
Table 5, it is clear that the
and bias of both estimators are decreasing with sample size. The decreasing
indicates that estimator predictions are approaching the true values with larger sample sizes, demonstrating enhanced accuracy and efficiency in estimation. The decreasing bias also shows the accuracy of the estimators.
The comparison of bias and between the presented estimators in the simulation for reveals that slightly outperforms in certain scenarios, particularly when dealing with heavy-tailed distribution and skewed distributions.