Estimation of Weighted Extropy Under the α-Mixing Dependence Condition

Abstract

Introduced as a complementary concept to Shannon entropy, extropy provides an alternative perspective for measuring uncertainty. While useful in areas such as reliability theory and scoring rules, extropy in its original form treats all outcomes equally, which can limit its applicability in real-world settings where different outcomes have varying degrees of importance. To address this, the weighted extropy measure incorporates a weight function that reflects the relative significance of outcomes, thereby increasing the flexibility and sensitivity of uncertainty quantification. In this paper, we propose a novel recursive non-parametric kernel estimator for weighted extropy based on $\alpha$-mixing dependent observations, a common setting in time series and stochastic processes. The recursive formulation allows for efficient updating with sequential data, making it particularly suitable for real-time analysis. We establish several theoretical properties of the estimator, including its recursive structure, consistency, and asymptotic behavior under mild regularity conditions. A comprehensive simulation study and data application demonstrate the practical performance of the estimator and validate its superiority over the non-recursive kernel estimator in terms of accuracy and computational efficiency. The results confirm the relevance of the method for dynamic, dependent, and weighted systems.

1. Introduction

Understanding and quantifying uncertainty is fundamental to many scientific disciplines. In this context, two pivotal measures—entropy and extropy—serve as core tools for characterizing the uncertainty associated with a random variable (RV). While entropy has been the foundation of information theory for decades, extropy offers a complementary perspective that has gained attention in recent years. This section introduces the basic definitions of these measures, highlights their extensions, and sets the stage for the motivation and objectives of the present study.

1.1. Definitions of Extropy and Its Extensions

Entropy was introduced by [1] as a basic measure of information. For a continuous RV X with support $(0,+\infty )$ and with probability density function (pdf) $f\left(x\right)$, the differential entropy associated with X is defined as follows:

$$H\left(X\right)=-\underset{0}{\overset{+\infty }{\int }}f\left(x\right)logf\left(x\right)dx.$$

(1)

This measure captures the average uncertainty inherent in the value of X and has found widespread application in areas such as communication systems, statistical inference, and thermodynamics [2,3]. For certain continuous distributions, entropy can potentially take negative values. For continuous distributions, Shannon’s entropy is not necessarily non-negative. In particular, when the pdf includes physical units, or under certain transformations, the computed entropy can take negative values.

Shannon’s entropy is a fundamental measure for quantifying the uncertainty of an $RV$, with broad applications in fields such as reliability, survival analysis, and actuarial science. Entropy is closely related to the logarithmic scoring rule, a widely used correct scoring method where the expected log score is the negative of the entropy. However, ref. [4] pointed out that entropy alone does not fully capture the information content of a distribution because the logarithmic score only considers the observed outcome, ignoring how the probability is distributed over the unobserved values. To overcome this limitation, they proposed to complement the logarithmic score with an additional scoring component, resulting in the total log score. This total score provides a more complete assessment because its expectation is equal to the sum of negentropy and negextropy, where negextropy is the negative of a newly introduced measure called extropy by [5].

Thus, the extropy proposed by [5] is a dual measure of entropy. In the same setting than the differential entropy, the differential extropy is given by

$$J\left(X\right)=-{\displaystyle \frac{1}{2}}\underset{0}{\overset{+\infty }{\int }}{f}^2\left(x\right)dx.$$

(2)

Both entropy and extropy are maximized by uniform distribution. For its properties and extensions, see [6].

Several extensions of entropy and extropy have been developed to adapt to specific inferential contexts. Ref. [7] observed that when it is known that a unit has survived to age t, entropy $H\left(X\right)$ no longer serves as an effective measure of uncertainty about the unit’s remaining lifetime. In such cases, the residual entropy function provides a more appropriate alternative. The residual extropy, introduced by [8], takes into account the uncertainty in the remaining lifetime of a system, given survival beyond a certain time t. It is defined by

$$J_t\left(X\right)=-{\displaystyle \frac{1}{2}}\underset{t}{\overset{+\infty }{\int }}{\left({\displaystyle \frac{f\left(x\right)}{\overline{F}\left(t\right)}}\right)}^{2}dx,$$

(3)

where $\overline{F}\left(t\right)=1-F\left(t\right)$ is the survival function of X.

Analogously, the past extropy accounts for the uncertainty if the failure occurs before time t. It is given by

$${\mathcal{J}}_t\left(X\right)=-{\displaystyle \frac{1}{2}}\underset{0}{\overset{t}{\int }}{\left({\displaystyle \frac{f\left(x\right)}{F\left(t\right)}}\right)}^{2}dx,$$

(4)

as discussed in [9,10]. These measures are particularly useful in survival and reliability analyses, where the timing of events carries crucial information.

A common limitation of these standard measures is their shift-invariance, as they depend only on the shape of the pdf $f\left(x\right)$, rather than on the values x themselves. To overcome this, weighted entropy was introduced by [11], as follows:

$$H^w\left(X\right)=-\underset{0}{\overset{+\infty }{\int }}xf\left(x\right)logf\left(x\right)dx.$$

(5)

This form integrates the influence of the values of the RV into the uncertainty quantification.

In parallel, the weighted extropy of [12] was defined. It is given by

$$J^w\left(X\right)=-{\displaystyle \frac{1}{2}}\underset{0}{\overset{+\infty }{\int }}x{f}^2\left(x\right)dx.$$

(6)

providing a refined view of concentration that incorporates magnitude. This measure can be rewritten using integration by parts. It is given as

$$\begin{array}{cc}\hfill J^w\left(X\right)& =-{\displaystyle \frac{1}{2}}\underset{0}{\overset{+\infty }{\int }}{f}^2\left(x\right)\underset{0}{\overset{x}{\int }}dydx\hfill \\ \hfill & =-{\displaystyle \frac{1}{2}}\underset{0}{\overset{+\infty }{\int }}dy\underset{y}{\overset{+\infty }{\int }}{f}^2\left(x\right)dx.\hfill \end{array}$$

(7)

They used illustrative examples to show that different distributions can have identical extropy values while having different weighted extropy values, and vice versa.

Recent developments have also introduced the notion of cumulative extropy. The weighted cumulative residual extropy (WCRJ), as defined in [13], is

$$WCRJ\left(X\right)=-{\displaystyle \frac{1}{2}}\underset{0}{\overset{+\infty }{\int }}x{\overline{F}}^2\left(x\right)dx,$$

while the weighted cumulative past extropy (WCPJ), discussed in [14], is

$$WCPJ\left(X\right)=-{\displaystyle \frac{1}{2}}\underset{0}{\overset{+\infty }{\int }}x{F}^2\left(x\right),dx.$$

These enhancements further enrich the tools for analyzing lifetime data and skewed distributions.

1.2. Applications and Estimation Context

Extropy and its extensions have been applied in various fields, including thermodynamics, astrophysics, and machine learning tasks such as speech recognition, as seen in [15,16,17,18]. While the original definition of extropy involves a single pdf, the extropy is particularly useful in comparing the uncertainties of two $RVs$ (see [19]). If $J\left(X_1\right)\le J\left(X_2\right)$, this suggests that $X_1$ is associated with greater uncertainty than $X_2$. In reliability theory, past and residual extropy are particularly useful for understanding systems under censoring or truncation, helping to quantify the information embedded in observed survival times.

A major drawback of the aforementioned information measures is their exclusive reliance on the pdf of the RV, ignoring the specific values it assumes. In particular, in the expressions presented in Equations (1) and (2), the integrand on the right-hand side is affected by x only through $f\left(x\right)$, resulting in a shift-invariant property. While this property may be mathematically convenient, it is often seen as a limitation in several practical applications, including mathematical neurobiology and reliability analysis. In such fields, accurate capture of uncertainty requires accounting for both the actual realizations of the RV and their corresponding probabilities. To overcome this, weighted versions of entropy and extropy have been proposed, which incorporate a weight function to emphasize the influence of particular realizations of the RV. These weighted measures provide a more nuanced characterization of uncertainty by reflecting the practical importance or criticality of particular values of the RV. From a statistical perspective, non-parametric estimation of extropy and its variants has gained attention. Ref. [20] employed extropy in the context of goodness-of-fit testing. Ref. [21] explored its estimation under progressively Type-II censored data, which is common in life-testing experiments. Recent work by [22,23] has introduced log-kernel estimators for extropy and weighted extropy, offering improved flexibility and robustness in reliability modeling and survival analysis.

However, most of these contributions assume independent and identically distributed (iid) data. This is problematic in some practical scenarios, such as those dealing with financial time series, physiological signals, or environmental data, where observations are temporally or spatially dependent. To account for such dependence, the framework of $\alpha$-mixing (or strong mixing) processes is often used.

Let us now give some mathematical background to the notion of $\alpha$-mixing. Consider a probability space $(\mathsf{\Omega },\mathcal{A},P)$ and let ${\mathcal{A}}_i^m$ denote the $\sigma$-algebra generated by the RVs $\{X_j;i\le j\le m\}$. A stationary process $\left\{X_j\right\}$ is said to fulfil the $\alpha$-mixing (strong mixing) condition if

$$\underset{B\in {\mathcal{A}}_{i+m}^{+\infty }}{\underset{A\in {\mathcal{A}}_{-\infty }^m}{sup}}|P(A\cap B)-P\left(A\right)P\left(B\right)|=\alpha \left(m\right)\to 0$$

as $m\to +\infty$. This implies that the dependence between the RVs $X_i$ and $X_{i+m}$ weakens as the separation m increases, leading to their asymptotic independence. The parameter $\alpha \left(m\right)$ is known as the mixing coefficient and represents the strength of the dependence.

Several recent studies have incorporated $\alpha$-mixing into the estimation of extropy-type measures. For instance, ref. [24] considered non-parametric estimation of past extropy in dependent contexts and [25] investigated negative cumulative extropy under $\alpha$-mixing. Ref. [26] introduced recursive kernel estimation methods for extropy under dependence, laying the groundwork for scalable computation. Despite these advances, only limited work has addressed the estimation of weighted extropy under dependence, especially in scenarios where data are received sequentially or in large volumes.

While traditional non-parametric techniques are robust and widely used in statistical analysis, they face significant difficulties when applied to data streams. Data streams are characterized by their continuous and rapid growth, facilitated by modern computing tools and data acquisition methods. As a result, practitioners are often faced with huge datasets that are too large for conventional storage, leading to numerous computational challenges.

Although the field of data streams is still relatively new, several methods have been developed to address the statistical issues involved. These include sequential approaches designed for successive experiments, which are particularly effective for non-parametric density estimation. Recursive estimators offer a significant computational advantage by allowing efficient updates with each incoming data point. Traditional non-recursive estimators can become computationally expensive in such settings. Recursive estimators provide an efficient alternative, allowing real-time updates as new data arrive without the need to recompute the entire dataset. This makes them particularly attractive for modern streaming data applications.

The primary objective of this study is to develop and analyze a recursive kernel estimator for weighted extropy under the $\alpha$-mixing dependence framework. Our proposed estimator is specifically tailored to capture both the probability structure and the magnitudes of observations in dependent settings. We explore its theoretical properties, including consistency and convergence rates, and assess its performance through detailed simulation studies. The proposed methodology aims to contribute a novel and practical tool for real-time uncertainty quantification in dependent data environments.

The work is explored in this way: In Section 2, we successfully develop a non-parametric recursive kernel estimator for the weighted extropy function. The recursive and asymptotic properties are established and discussed in Section 3. Section 4 presents the evaluation of the performance of the estimator using both simulated and real data, along with a comparative analysis of the recursive and non-recursive weighted extropy estimators. The concluding remarks are given in Section 5.

2. Non-Parametric Estimation of Weighted Extropy Under $\mathit{\alpha }$-Mixing Dependence

In this section, we address the problem of estimating the weighted extropy of a dependent sequence of RVs, where the dependence is governed by the $\alpha$-mixing condition. The motivation arises from the need to construct a recursive, non-parametric estimator of the weighted extropy that can be efficiently updated as new data become available, making it suitable for real-time or sequential analysis. This is particularly important in practical applications where observations are not independent but exhibit dependence over time, such as in financial markets, environmental monitoring, or reliability assessment. Existing methods for weighted extropy estimation generally assume independence and often rely on non-recursive techniques, limiting their applicability in dependent environments (see [23]).

Let $\{X_j;1\le j\le m\}$ be a sequence of identically distributed RVs representing the lifetimes of m components. These variables are assumed to have $\alpha$-mixing dependence, a broad class of dependence structures that includes many practical time series and stochastic models.

The objective is to estimate the weighted extropy, defined in Equation (6). To address this, we consider the non-parametric recursive kernel-based estimator, which incorporates dependence through the use of $\alpha$-mixing conditions.

We start with the non-parametric recursive kernel estimator of the pdf proposed by [27], as follows:

$${\widehat{f}}_m\left(x\right)={\displaystyle \frac{1}{m}}\sum _{j=1}^m{\displaystyle \frac{1}{{\rho }_j}}\mathcal{K}\left({\displaystyle \frac{x-X_j}{{\rho }_j}}\right),$$

(8)

where $\mathcal{K}\left(x\right)$ is a bounded, symmetric, non-negative kernel function satisfying

$$\underset{-\infty }{\overset{+\infty }{\int }}\mathcal{K}\left(x\right)dx=1,\underset{-\infty }{\overset{+\infty }{\int }}x\mathcal{K}\left(x\right)dx=0,$$

and $\left\{{\rho }_m\right\}$ is a sequence of bandwidths satisfying ${\rho }_m\to 0$ and $m{\rho }_m\to +\infty$ as $m\to +\infty$. In addition, the kernel and bandwidths satisfy the regularity conditions introduced by [28] to handle $\alpha$-mixing data.

This estimator can be updated recursively as follows:

$${\widehat{f}}_m\left(x\right)={\displaystyle \frac{m-1}{m}}{\widehat{f}}_{m-1}\left(x\right)+{\displaystyle \frac{1}{m{\rho }_m}}\mathcal{K}\left({\displaystyle \frac{x-X_m}{{\rho }_m}}\right),$$

(9)

making it particularly useful for sequential data analysis.

Under the $\alpha$-mixing assumption, the asymptotic properties of ${\widehat{f}}_m\left(x\right)$ are given by the following approximations for the bias and variance:

$$\mathrm{Bias}\left({\widehat{f}}_m\left(x\right)\right)\backsimeq {\displaystyle \frac{{\rho }_m^s{\mathfrak{C}}_s}{s!}}{f}^{\left(s\right)}\left(x\right){\gamma }_s$$

(10)

and

$$\mathrm{Var}\left({\widehat{f}}_m\left(x\right)\right)\backsimeq {\displaystyle \frac{{\theta }_{1}f\left(x\right)}{m{\rho }_m}}{\mathfrak{C}}_{\mathcal{K}},$$

(11)

respectively, where ${\mathfrak{C}}_s=\int u^s\mathcal{K}\left(u\right)du$ and ${\mathfrak{C}}_{\mathcal{K}}=\int {\mathcal{K}}^2\left(u\right)du$.

Recursive Estimator for Weighted Extropy

Based on the recursive pdf estimator in Equation (8), the proposed estimator for the weighted extropy is

$${\widehat{\mathcal{J}}}_m^w\left(X\right)=-{\displaystyle \frac{1}{2}}\underset{0}{\overset{+\infty }{\int }}x{\widehat{f}}_m^2\left(x\right)dx.$$

(12)

An alternative equivalent form is

$${\widehat{\mathcal{J}}}_m^w\left(X\right)=-{\displaystyle \frac{1}{2}}\underset{0}{\overset{+\infty }{\int }}dy\underset{y}{\overset{+\infty }{\int }}{\widehat{f}}_m^2\left(x\right)dx,$$

which may offer computational advantages in some contexts.

3. Recursive and Asymptotic Properties

In this section, we establish two key properties of the proposed estimator ${\widehat{\mathcal{J}}}_m^w\left(X\right)$ for the weighted extropy under $\alpha$-mixing dependence: its recursive structure and its asymptotic behavior, including consistency and asymptotic normality. These results confirm the usefulness and theoretical soundness of the estimator introduced in Section 2. The asymptotic properties of the estimator are derived using the assumptions presented by [28,29]. Throughout the paper, we will use the notation $f\left(x\right)\backsimeq g\left(x\right)$ to denote $f\left(x\right)=g\left(x\right)(1+o(1\left)\right)$.

A significant advantage of the proposed estimator is its recursive nature, which allows efficient computation in online or sequential settings without the need to recompute the entire estimator as new observations become available. The following lemma formally states the recursive form of ${\widehat{\mathcal{J}}}_m^w\left(X\right)$.

Lemma 1.

Let ${\widehat{\mathcal{J}}}_m^w\left(X\right)$ be the non-parametric recursive estimator of the weighted extropy $J^w\left(X\right)$, defined in Equation (12), and assume that the regularity conditions stated in Section 2 are satisfied. Then, the estimator satisfies the following recursive relation:

$$\begin{array}{c}\hfill {\widehat{\mathcal{J}}}_m^w\left(X\right)={\displaystyle \frac{{(m-1)}^2}{m^2}}{\widehat{\mathcal{J}}}_{m-1}^w\left(X\right)-{\displaystyle \frac{1}{m^2{\rho }_m}}\left\{\underset{0}{\overset{+\infty }{\int }}x\mathcal{K}\left({\displaystyle \frac{x-X_m}{{\rho }_m}}\right)((m-1){\widehat{f}}_{m-1}\left(x\right)\right.\\ \hfill \left.+{\displaystyle \frac{1}{2{\rho }_m}}\mathcal{K}\left({\displaystyle \frac{x-X_m}{{\rho }_m}}\right))dx\right\}.\end{array}$$

(13)

Proof. 

We have

$${\widehat{\mathcal{J}}}_m^w\left(X\right)={\displaystyle \frac{-1}{2}}\underset{0}{\overset{+\infty }{\int }}x{\widehat{f}}_m^2\left(x\right)dx,$$

(14)

$${\widehat{\mathcal{J}}}_{m-1}^w\left(X\right)={\displaystyle \frac{-1}{2}}\underset{0}{\overset{+\infty }{\int }}x{\widehat{f}}_{m-1}^2\left(x\right)dx$$

(15)

and

$${\widehat{f}}_m\left(x\right)={\displaystyle \frac{m-1}{m}}{\widehat{f}}_{m-1}\left(x\right)+{\displaystyle \frac{1}{m{\rho }_m}}\mathcal{K}\left({\displaystyle \frac{x-X_m}{{\rho }_m}}\right).$$

(16)

So, we obtain

$$\begin{array}{cc}\hfill -2{\widehat{\mathcal{J}}}_m^w\left(X\right)& =\underset{0}{\overset{+\infty }{\int }}x{\widehat{f}}_m^2\left(x\right)dx\hfill \\ \hfill & =\underset{0}{\overset{+\infty }{\int }}x{\left[{\displaystyle \frac{m-1}{m}}{\widehat{f}}_{m-1}\left(x\right)+{\displaystyle \frac{1}{m{\rho }_m}}\mathcal{K}\left({\displaystyle \frac{x-X_m}{{\rho }_m}}\right)\right]}^{2}dx.\hfill \end{array}$$

(17)

Expanding, we obtain

$$\begin{array}{cc}\hfill -2{\widehat{\mathcal{J}}}_m^w\left(X\right)=& {\displaystyle \frac{{(m-1)}^2}{m^2}}\underset{0}{\overset{+\infty }{\int }}x{\widehat{f}}_{m-1}^2\left(x\right)dx+{\displaystyle \frac{1}{m^2{\rho }_m^2}}\underset{0}{\overset{+\infty }{\int }}x{\mathcal{K}}^2\left({\displaystyle \frac{x-X_m}{{\rho }_m}}\right)dx\hfill \\ \hfill & +{\displaystyle \frac{2(m-1)}{m^2{\rho }_m}}\underset{0}{\overset{+\infty }{\int }}x{\widehat{f}}_{m-1}\left(x\right)\mathcal{K}\left({\displaystyle \frac{x-X_m}{{\rho }_m}}\right)dx\hfill \\ \hfill =& {\displaystyle \frac{{(m-1)}^2}{m^2}}\left(-2{\widehat{\mathcal{J}}}_{m-1}^w\left(X\right)\right)+{\displaystyle \frac{1}{{\left(m{\rho }_m\right)}^2}}\underset{0}{\overset{+\infty }{\int }}x{\mathcal{K}}^2\left({\displaystyle \frac{x-X_m}{{\rho }_m}}\right)dx\hfill \\ \hfill & +{\displaystyle \frac{2(m-1)}{m^2{\rho }_m}}\underset{0}{\overset{+\infty }{\int }}x{\widehat{f}}_{m-1}\left(x\right)\mathcal{K}\left({\displaystyle \frac{x-X_m}{{\rho }_m}}\right)dx.\hfill \end{array}$$

(18)

So, we have

$$\begin{array}{cc}\hfill {\widehat{\mathcal{J}}}_m^w\left(X\right)=& {\displaystyle \frac{{(m-1)}^2}{m^2}}{\widehat{\mathcal{J}}}_{m-1}^w\left(X\right)-{\displaystyle \frac{1}{m^2{\rho }_m}}\left\{\underset{0}{\overset{+\infty }{\int }}x\mathcal{K}\left({\displaystyle \frac{x-X_m}{{\rho }_m}}\right)((m-1){\widehat{f}}_{m-1}\left(x\right)\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{1}{2{\rho }_m}}\mathcal{K}\left({\displaystyle \frac{x-X_m}{{\rho }_m}}\right))dx\right\}.\hfill \end{array}$$

(19)

The theorem is thereby established. □

This result allows the estimator to be efficiently updated at each step, making it particularly useful in dynamic environments where observations arrive sequentially and recomputation would be computationally intensive.

Next, we establish the consistency of the estimator ${\widehat{\mathcal{J}}}_m^w\left(X\right)$. This ensures that as the sample size increases, the estimator is likely to converge to the true value of the weighted extropy $J^w\left(X\right)$.

Theorem 1.

Let ${\widehat{\mathcal{J}}}_m^w\left(X\right)$ be defined as in Equation (12), and assume that the regularity conditions outlined in Section 2 hold. Then, the estimator is weakly consistent, i.e.,

$${\widehat{\mathcal{J}}}_m^w\left(X\right)\stackrel{p}{⟶}{J}^w\left(X\right).$$

Proof. 

Applying a Taylor series expansion to ${\widehat{f}}_m^2\left(x\right)$, we obtain

$${\widehat{f}}_m^2\left(x\right)\backsimeq f^2\left(x\right)+2\left({\widehat{f}}_m\left(x\right)-f\left(x\right)\right)f\left(x\right).$$

(20)

Then, we have

$$\begin{array}{cc}\hfill \underset{y}{\overset{+\infty }{\int }}{\widehat{f}}_m^2\left(x\right)dx\backsimeq & \underset{y}{\overset{+\infty }{\int }}{f}^2\left(x\right)dx+2\underset{y}{\overset{+\infty }{\int }}\left({\widehat{f}}_m\left(x\right)-f\left(x\right)\right)f\left(x\right)dx.\hfill \\ \hfill -{\displaystyle \frac{1}{2}}\underset{0}{\overset{+\infty }{\int }}dy\underset{y}{\overset{+\infty }{\int }}{\widehat{f}}_m^2\left(x\right)dx\backsimeq & -{\displaystyle \frac{1}{2}}\underset{0}{\overset{+\infty }{\int }}dy\underset{y}{\overset{+\infty }{\int }}{f}^2\left(x\right)dx-\underset{0}{\overset{+\infty }{\int }}dy\underset{y}{\overset{+\infty }{\int }}\left({\widehat{f}}_m\left(x\right)-f\left(x\right)\right)f\left(x\right)dx.\hfill \end{array}$$

(21)

As a result, we derive

$${\widehat{\mathcal{J}}}_m^w\left(X\right)\backsimeq J^w\left(X\right)-\underset{0}{\overset{+\infty }{\int }}dy\underset{y}{\overset{+\infty }{\int }}\left({\widehat{f}}_m\left(x\right)-f\left(x\right)\right)f\left(x\right)dx.$$

(22)

Consequently, applying Equation (10), the bias of ${\widehat{\mathcal{J}}}_m^w\left(X\right)$ can be expressed as

$$\begin{array}{cc}\hfill \mathrm{Bias}\left({\widehat{\mathcal{J}}}_m^w\left(X\right)\right)\backsimeq & -\underset{0}{\overset{+\infty }{\int }}dy\underset{y}{\overset{+\infty }{\int }}\mathrm{Bias}\left(f_m\left(x\right)\right)f\left(x\right)dx\hfill \\ \hfill \backsimeq & -{\displaystyle \frac{{\rho }_m^s{\gamma }_s{\mathfrak{C}}_s}{s!}}\underset{0}{\overset{+\infty }{\int }}dy\underset{y}{\overset{+\infty }{\int }}{f}^{\left(s\right)}\left(x\right)f\left(x\right)dx.\hfill \end{array}$$

Using Equation (11), the variance of ${\widehat{\mathcal{J}}}_m^w\left(X\right)$ is obtained as

$$\begin{array}{cc}\hfill \mathrm{Var}\left({\widehat{\mathcal{J}}}_m^w\left(X\right)\right)\backsimeq & \underset{0}{\overset{+\infty }{\int }}dy\underset{y}{\overset{+\infty }{\int }}\mathrm{Var}\left(f_m\left(x\right)\right)f^2\left(x\right)dx\hfill \\ \hfill \backsimeq & {\displaystyle \frac{{\theta }_1}{m{\rho }_m}}{\mathfrak{C}}_{\mathcal{K}}\underset{0}{\overset{+\infty }{\int }}dy\underset{y}{\overset{+\infty }{\int }}{f}^3\left(x\right)dx.\hfill \end{array}$$

Then, we have

$$\begin{array}{cc}\hfill MSE\left({\widehat{\mathcal{J}}}_m^w\left(X\right)\right)=& \mathrm{Bias}{\left({\widehat{\mathcal{J}}}_m^w\left(X\right)\right)}^2+\mathrm{Var}\left({\widehat{\mathcal{J}}}_m^w\left(X\right)\right)\hfill \\ \hfill & \backsimeq {\left(-{\displaystyle \frac{{\rho }_m^s{\gamma }_s{\mathfrak{C}}_s}{s!}}\underset{0}{\overset{+\infty }{\int }}dy\underset{y}{\overset{+\infty }{\int }}{f}^{\left(s\right)}\left(x\right)f\left(x\right)dx\right)}^2+{\displaystyle \frac{{\theta }_1}{m{\rho }_m}}{\mathfrak{C}}_{\mathcal{K}}\underset{0}{\overset{+\infty }{\int }}dy\underset{y}{\overset{+\infty }{\int }}{f}^3\left(x\right)dx.\hfill \end{array}$$

When m approaches to $+\infty$, we find that

$$MSE\left({\widehat{\mathcal{J}}}_m^w\left(X\right)\right)\to 0,$$

(23)

Hence, we obtain

$${\widehat{\mathcal{J}}}_m^w\left(X\right)\stackrel{p}{\to }{J}^w\left(X\right).$$

This ends the proof. □

Beyond consistency, we also investigate the asymptotic distribution of the proposed estimator. The next theorem demonstrates that ${\widehat{\mathcal{J}}}_m^w\left(X\right)$ is asymptotically normal, allowing for inference procedures such as confidence intervals and hypothesis testing.

Theorem 2.

Assume that ${\widehat{\mathcal{J}}}_m^w\left(X\right)$ is the non-parametric recursive estimator defined in Equation (12), and that the conditions specified in Section 2 are satisfied. Then,

$${\left(m{\rho }_m\right)}^{{\displaystyle \frac{1}{2}}}\left\{{\displaystyle \frac{{\widehat{\mathcal{J}}}_m^w\left(X\right)-J^w\left(X\right)}{{\psi }_J}}\right\}$$

(24)

follows the Gaussian $N(0,1)$ distribution when m tends to $+\infty$ with

$${\psi }_J^2\backsimeq {\theta }_1{\mathfrak{C}}_{\mathcal{K}}\underset{0}{\overset{+\infty }{\int }}dy\underset{y}{\overset{+\infty }{\int }}{f}^3\left(x\right)dx.$$

(25)

Proof. 

Using Equation (22), we have

$${\left(m{\rho }_m\right)}^{{\displaystyle \frac{1}{2}}}\left({\widehat{\mathcal{J}}}_m^w\left(X\right)-J^w\left(X\right)\right)\backsimeq -{\left(m{\rho }_m\right)}^{{\displaystyle \frac{1}{2}}}\underset{0}{\overset{+\infty }{\int }}dy\underset{y}{\overset{+\infty }{\int }}({\widehat{f}}_m\left(x\right)-f\left(x\right))f\left(x\right)dx.$$

(26)

Under the assumption of asymptotic normality for ${\widehat{f}}_m\left(x\right)$, as demonstrated by [28], it can be inferred that, as m approaches $+\infty$,

$${\left(m{\rho }_m\right)}^{{\displaystyle \frac{1}{2}}}\left\{{\displaystyle \frac{{\widehat{\mathcal{J}}}_m^w\left(X\right)-J^w\left(X\right)}{{\psi }_J}}\right\}$$

has the Gaussian $N(0,1)$ distribution with ${\psi }_J^2$ presented in Equation (25). This ends the proof. □

The derivation relies on the asymptotic normality of the recursive kernel estimator ${\widehat{f}}_m\left(x\right)$ under $\alpha$-mixing dependence.

Together, these results provide a strong theoretical foundation for the proposed estimator. The recursive structure ensures computational tractability, while consistency and asymptotic normality provide the basis for statistical inference. Compared to the traditional estimators that assume independence, our method accommodates dependent data, offering a more general and practically relevant approach to weighted extropy estimation.

This formulation generalizes classical kernel estimation techniques and extends existing weighted extropy-based methods to more realistic data-generating mechanisms. The main novelty of this work is the proposal of a recursive kernel estimator for the weighted extropy under $\alpha$-mixing conditions, which extends existing methods developed only for iid data. Unlike the non-recursive estimators, the recursive approach allows efficient real-time updating of the estimates in dependent data settings. A detailed comparison shows that the proposed estimator performs competitively and offers practical advantages. This work thus advances weighted extropy estimation by addressing dependence structures that have not been fully explored in the previous literature.

In later sections, we conduct simulation studies to evaluate the finite-sample performance of the proposed estimator under various $\alpha$-mixing models. These results demonstrate the effectiveness, convergence behavior and performance improvements of the estimator over non-recursive or independence assumption methods.

4. Numerical Illustration

In this section, we reflect on the suitability of the proposed estimator using a simulation study and real data.

4.1. Simulation

In order to assess the performance of the estimator outlined in Equation (12), we conduct a comprehensive simulation study using both Gaussian and exponential distributions. In the case of the Gaussian distribution, the parameters are chosen as $\mu =-1$ and $\sigma =6$, whereas for the exponential distribution, the parameter is set to $\lambda =10$. The data for each distribution are generated using an autoregressive of order 1 ($AR\left(1\right)$) process with a correlation coefficient $\varphi =0.5$ to ensure that the simulated data exhibit the serial dependence common to time series data.

The kernel functions selected for this simulation study are the Gaussian and Epanechnikov kernels, which are standard choices for kernel density estimation due to their smoothness and computational efficiency. The Epanechnikov kernel is a widely used kernel in non-parametric density estimation due to its optimality in minimizing the mean integrated squared error (see [30]). It is defined as

$$K\left(u\right)={\displaystyle \frac{3}{4}}(1-u^2){\mathbf{1}}_{\left\{\right|u|\le 1\}},$$

where ${\mathbf{1}}_{\left\{\right|u|\le 1\}}$ is the indicator function. The kernel is symmetric, has compact support, and gives more weight to observations closer to the target point, making it both efficient and computationally attractive. Following [28], we take the bandwidth values ${\rho }_i$ as $\{i^{-\nu };i=1,2,\cdots ,n\}$, with $0<\nu <1$.

The performance of the proposed estimator is assessed for a range of sample sizes to determine how the estimator behaves as the data size increases. For each sample size, two key metrics are calculated: the absolute bias ($\left|bias\right|$) and the mean squared error ($MSE$). These metrics give an indication of the accuracy and consistency of the estimator. The $\left|bias\right|$ reflects the average deviation of the estimator from the true value, while the $MSE$ provides an overall measure of the performance of the estimator. By comparing these metrics across the Gaussian and exponential distributions, we can determine how well the estimator performs in different settings. This simulation study is crucial for understanding the practical performance of the estimator under realistic conditions.

To facilitate a meaningful comparison between the recursive and non-recursive estimators for weighted extropy, we introduce a non-recursive estimator as a benchmark. The non-recursive estimator is derived using standard kernel density estimation techniques, and its expression is given by

$${\widehat{J}}_m^{\ast w}\left(X\right)=-{\displaystyle \frac{1}{2}}\underset{0}{\overset{+\infty }{\int }}x{\widehat{f}}_m^{\ast 2}\left(x\right)dx,$$

(27)

where the kernel density estimator, ${\widehat{f}}_m^{\ast }\left(x\right)$, is formulated as follows:

$${\widehat{f}}_m^{\ast }\left(x\right)={\displaystyle \frac{1}{m{\rho }_m}}\sum _{j=1}^m\mathcal{K}\left({\displaystyle \frac{x-X_j}{{\rho }_m}}\right),$$

(28)

with $\mathcal{K}\left(x\right)$ representing the kernel function and ${\rho }_m$ being a sequence of bandwidths satisfying the conditions ${\rho }_m\to 0$ and $m{\rho }_m\to +\infty$ as m increases. This non-recursive approach is one of the most commonly used techniques for density estimation, as introduced by [31], and is considered a standard in the literature.

It is important to note that the non-recursive estimator of the weighted extropy serves as a critical benchmark for assessing the performance of the proposed estimator.

In this context, the recursive estimator of the weighted extropy is expected to offer improved performance by incorporating recursive relationships within the data, leading to more accurate and stable estimates of the weighted extropy, especially under the dependence structure and different sample sizes.

In order to empirically assess the performance and comparison of both estimators, we conduct a comprehensive simulation study. The simulation setup described earlier, based on Gaussian and exponential distributions under an AR(1) process with $\varphi =0.5$, is also used here to evaluate the estimator in Equation (27).

In the simulation study, different sample sizes are considered, and for each, the estimated weighted extropy, $\left|bias\right|$, and the $MSE$ are calculated.

In addition, two different values of the bandwidth parameter, ${\rho }_m=0.5$ and ${\rho }_m=0.7$, are considered specifically for the non-recursive estimator. The results of the simulation study, which include the performance comparison between the recursive and non-recursive estimators of the weighted extropy using the two different kernel functions, are summarized in

Table 1. Simulation results for ${\widehat{\mathcal{J}}}_m^w\left(X\right)$ and ${\widehat{J}}_m^{\ast w}\left(X\right)$ under the exponential AR(1) model with $J^w\left(X\right)=-0.125$, using the Gaussian kernel, including the estimates, $\left|bias\right|$ and $MSE$.

m	${\widehat{\mathcal{J}}}_{\mathit{m}}^{\mathit{w}}\left(\mathit{X}\right)$	$\left|\mathit{bias}\right|$	$\mathit{MSE}$	${\widehat{\mathit{J}}}_{\mathit{m}}^{\ast \mathit{w}}\left(\mathit{X}\right)$	$\left|\mathit{bias}\right|$	$\mathit{MSE}$	${\widehat{\mathit{J}}}_{\mathit{m}}^{\ast \mathit{w}}\left(\mathit{X}\right)$	$\left|\mathit{bias}\right|$	$\mathit{MSE}$
				($\mathit{\rho }=\mathbf{0.5}$)			($\mathit{\rho }=\mathbf{0.7}$)
50	−0.0672	0.0578	0.0034	−0.05447	0.07053	0.00499	−0.05033	0.07467	0.00558
ine 100	−0.0768	0.0482	0.0024	−0.05455	0.07045	0.00497	−0.05037	0.07463	0.00557
150	−0.0824	0.0426	0.0019	−0.05476	0.07024	0.00494	−0.05038	0.07462	0.00556
200	−0.0866	0.0384	0.0015	−0.05474	0.07026	0.00494	−0.05039	0.07461	0.00551
250	−0.0897	0.0353	0.0013	−0.05473	0.07027	0.00494	−0.05043	0.07457	0.00507
300	−0.0924	0.0326	0.0011	−0.05475	0.07025	0.00494	−0.05044	0.07456	0.00481
350	−0.0943	0.0307	0.0010	−0.05479	0.07021	0.00493	−0.05046	0.07454	0.00480
400	−0.0963	0.0287	0.0009	−0.05474	0.07026	0.00494	−0.05047	0.07453	0.00476
450	−0.0977	0.0273	0.0008	−0.05473	0.07027	0.00494	−0.05049	0.07451	0.00475
500	−0.0989	0.0261	0.0007	−0.05476	0.07024	0.00494	−0.05049	0.07450	0.00475

,

Table 2. Simulation results for ${\widehat{\mathcal{J}}}_m^w\left(X\right)$ and ${\widehat{J}}_m^{\ast w}\left(X\right)$ under the exponential AR(1) model with $J^w\left(X\right)=-0.125$, using the Epanechnikov kernel, including the $\left|bias\right|$ and $MSE$.

m	${\widehat{\mathcal{J}}}_{\mathit{m}}^{\mathit{w}}\left(\mathit{X}\right)$	$\left|\mathit{bias}\right|$	$\mathit{MSE}$	${\widehat{\mathit{J}}}_{\mathit{m}}^{\ast \mathit{w}}\left(\mathit{X}\right)$	$\left|\mathit{b}\mathit{i}\mathit{a}\mathit{s}\right|$	$\mathit{MSE}$	${\widehat{\mathit{J}}}_{\mathit{m}}^{\ast \mathit{w}}\left(\mathit{X}\right)$	$\left|\mathit{bias}\right|$	$\mathit{MSE}$
				($\mathit{\rho }=\mathbf{0.5}$)			($\mathit{\rho }=\mathbf{0.7}$)
50	−0.09138	0.03362	0.00131	−0.07475	0.05025	0.00257	−0.06741	0.05759	0.00334
100	−0.10085	0.02415	0.00073	−0.07484	0.05016	0.00253	−0.06749	0.05751	0.00332
150	−0.10487	0.02013	0.00051	−0.07492	0.05008	0.00251	−0.06765	0.05735	0.00331
200	−0.10734	0.01766	0.00040	−0.07493	0.05007	0.00250	−0.06760	0.05740	0.00331
250	−0.11009	0.01491	0.00029	−0.07495	0.05005	0.00250	−0.06758	0.05742	0.00330
300	−0.11160	0.01340	0.00025	−0.07497	0.05003	0.00249	−0.06762	0.05738	0.00330
350	−0.11236	0.01264	0.00021	−0.07502	0.04998	0.00248	−0.06763	0.05737	0.00329
400	−0.11342	0.01158	0.00019	−0.07509	0.04991	0.00248	−0.06765	0.05735	0.00329
450	−0.11443	0.01057	0.00016	−0.07512	0.04989	0.00247	−0.06770	0.05730	0.00329
500	−0.11466	0.01034	0.00015	−0.07511	0.04989	0.00246	−0.06775	0.05725	0.00328

,

Table 3. Simulation results for ${\widehat{\mathcal{J}}}_m^w\left(X\right)$ and ${\widehat{J}}_m^{\ast w}\left(X\right)$ under the Gaussian AR(1) model with $J^w\left(X\right)=-0.03044$, using the Gaussian kernel, including the estimates, $\left|bias\right|$ and $MSE$.

m	${\widehat{\mathcal{J}}}_{\mathit{m}}^{\mathit{w}}\left(\mathit{X}\right)$	$\left|\mathit{bias}\right|$	$\mathit{MSE}$	${\widehat{\mathit{J}}}_{\mathit{m}}^{\ast \mathit{w}}\left(\mathit{X}\right)$	$\left|\mathit{bias}\right|$	$\mathit{MSE}$	${\widehat{\mathit{J}}}_{\mathit{m}}^{\ast \mathit{w}}\left(\mathit{X}\right)$	$\left|\mathit{bias}\right|$	$\mathit{MSE}$
				($\mathit{\rho }=\mathbf{0.5}$)			($\mathit{\rho }=\mathbf{0.7}$)
50	−0.03875	0.00832	0.000452	−0.01078	0.01965	0.00039	−0.00994	0.02059	0.00043
100	−0.03318	0.00274	0.000155	−0.01082	0.01962	0.00039	−0.00988	0.02055	0.00043
150	−0.02857	0.00187	0.000093	−0.01092	0.01951	0.00039	−0.01000	0.02043	0.00042
200	−0.03167	0.00123	0.000061	−0.01112	0.01932	0.00038	−0.01025	0.02019	0.00042
250	−0.02922	0.00121	0.000057	−0.01121	0.01923	0.00038	−0.01025	0.02018	0.00042
300	−0.02939	0.00105	0.000044	−0.01128	0.01915	0.00038	−0.01026	0.02017	0.00041
350	−0.02947	0.00096	0.000037	−0.01172	0.01871	0.00037	−0.01041	0.02003	0.00041
400	−0.02995	0.00049	0.000035	−0.01209	0.01834	0.00036	−0.01065	0.01979	0.00041
450	−0.02996	0.00048	0.000031	−0.01249	0.01794	0.00036	−0.01098	0.01946	0.00041
500	−0.03024	0.00019	0.000028	−0.01464	0.01580	0.00033	−0.01180	0.01864	0.00040

and

Table 4. Simulation results for ${\widehat{\mathcal{J}}}_m^w\left(X\right)$ and ${\widehat{J}}_m^{\ast w}\left(X\right)$ under the Gaussian AR(1) model with $J^w\left(X\right)=-0.03044$, using the Epanechnikov kernel, including the estimates, $\left|bias\right|$ and $MSEs$.

m	${\widehat{\mathcal{J}}}_{\mathit{m}}^{\mathit{w}}\left(\mathit{X}\right)$	$\left|\mathit{bias}\right|$	$\mathit{MSE}$	${\widehat{\mathit{J}}}_{\mathit{m}}^{\ast \mathit{w}}\left(\mathit{X}\right)$	$\left|\mathit{bias}\right|$	$\mathit{MSE}$	${\widehat{\mathit{J}}}_{\mathit{m}}^{\ast \mathit{w}}\left(\mathit{X}\right)$	$\left|\mathit{bias}\right|$	$\mathit{MSE}$
				($\mathit{\rho }=\mathbf{0.5}$)			($\mathit{\rho }=\mathbf{0.7}$)
50	−0.03740	0.00697	0.000346	−0.05136	0.02093	0.00104	−0.01093	0.01951	0.00039
100	−0.03283	0.00240	0.000136	−0.04362	0.01318	0.00040	−0.01095	0.01949	0.00039
150	−0.03161	0.00118	0.000095	−0.04096	0.01053	0.00026	−0.01100	0.01943	0.00039
200	−0.03146	0.00103	0.000063	−0.03885	0.00842	0.00017	−0.01129	0.01914	0.00038
250	−0.03113	0.00070	0.000054	−0.03789	0.00745	0.00013	−0.01152	0.01892	0.00037
300	−0.03111	0.00067	0.000044	−0.03720	0.00676	0.00011	−0.01172	0.01872	0.00036
350	−0.03080	0.00037	0.000034	−0.03684	0.00641	0.00009	−0.01213	0.01831	0.00035
400	−0.03072	0.00028	0.000032	−0.03615	0.00572	0.00008	−0.01231	0.01813	0.00035
450	−0.03071	0.00028	0.000026	−0.03552	0.00508	0.00007	−0.01316	0.01728	0.00034
500	−0.03040	0.00003	0.000025	−0.03519	0.00475	0.00006	−0.01478	0.01566	0.00033

. These tables present the detailed performance metrics, such as the estimated value, $\left|bias\right|$, and $MSE$ for both estimators under different sample sizes.

Analyzing the results presented in the tables, it is evident that both the $MSE$ and $\left|bias\right|$ associated with the recursive estimator ${\widehat{\mathcal{J}}}_m^w\left(X\right)$ show a consistent decreasing trend with increasing sample size, regardless of the kernel function used. This pattern reflects the increasing accuracy and reliability of the estimator, suggesting that larger samples lead to more stable and precise approximations. The reduction in $MSE$ suggests that the recursive estimator becomes more efficient with increasing data, while the decreasing bias highlights the improved closeness of the estimator to the true value of the weighted extropy.

Similarly, the non-recursive estimator ${\widehat{J}}_m^{\ast w}\left(X\right)$ also shows a decreasing trend in both $\left|bias\right|$ and $MSE$ as the sample size increases, confirming the general principle that larger samples improve estimation performance. However, across all evaluated scenarios, the recursive estimator ${\widehat{\mathcal{J}}}_m^w\left(X\right)$ consistently outperforms its non-recursive counterpart. This superiority is in line with the expected theoretical advantages of recursive estimation, especially in dealing with dependent data more efficiently and adaptively.

Furthermore, the comparative analysis of the two kernel functions, the Gaussian and the Epanechnikov kernel, shows that the Epanechnikov kernel consistently yields lower $MSE$ values in both the exponential and Gaussian AR(1) processes. This advantage holds for both estimators, reinforcing the utility of the Epanechnikov kernel in reducing estimation error and improving performance. The results underscore the importance of kernel selection and sample size in refining estimator behavior and ultimately achieving more accurate estimates of weighted extropy in dependent data contexts.

4.2. Data Analysis

• Data 1

The practical applicability of our proposed estimator is validated through the analysis of real-world data. In this study, we utilize the dataset presented in [32]. The failure times of 18 electronic devices were analyzed, and an exponential distribution with parameter $\lambda =0.00581$ was fitted to the data. The Kolmogorov–Smirnov statistic was calculated as 0.1249, with a corresponding p-value of 0.9091, indicating that the exponential distribution is a suitable choice for modeling the data.

To evaluate the performance of the estimators of weighted extropy defined in Equations (12) and (27), we employed 12 bootstrap samples, each of size 18. The corresponding estimate, $\left|bias\right|$, and $MSE$ were computed. The Epanechnikov kernel function was applied for the data analysis part as it gave the lowest $MSE$ for both estimators in both the exponential and Gaussian AR(1) distribution scenarios. The results are shown in

Table 5. Estimated value, bias, $MSE$ of ${\widehat{\mathcal{J}}}_m^w\left(X\right)$ and ${\widehat{J}}_m^{\ast w}\left(X\right)$ with $J^w\left(X\right)=-0.125$ for the real data.

Estimator		Bias	$\mathit{MSE}$
${\widehat{\mathcal{J}}}_m^w\left(X\right)$	−0.0343	0.0907	0.0104
${\widehat{J}}_m^{\ast w}\left(X\right)$	−0.0216	0.1034	0.0115

.

As can be seen from Table 5, the recursive kernel estimator ${\widehat{\mathcal{J}}}_m^w\left(X\right)$ outperforms the non-recursive estimator ${\widehat{J}}_m^{\ast w}\left(X\right)$.

• Data 2

In this section, we perform a detailed statistical analysis of the Nile River annual flow data, a well-known univariate time series representing the yearly flow (in ${10}^8$ m³) of the Nile River at Aswan from 1871 to 1970. The annual flow data of the Nile River were obtained from the $datasets$ package in R, which provides access to publicly available historical hydrological records. The dataset exhibits features commonly observed in hydrological time series, such as autocorrelation and possible regime changes. Our analysis proceeds through model fitting using an AR(1) structure, and residual diagnostics to assess the adequacy of the model.

The time series plot of the data is given in Figure 1.

To assess dependence over time, we plot the autocorrelation function (ACF) and partial autocorrelation function (PACF) of the series that are given in Figure 2 and Figure 3.

The ACF gradually tails off, and the PACF cuts off sharply after lag 1, indicating that an AR(1) model may be suitable for the data.

We fit an AR(1) model to the Nile data. The estimated coefficient is ${\widehat{\varphi }}_1=0.5063$, and the estimated innovation variance is ${\widehat{\sigma }}^2=21125$.

We examine the residuals from the AR(1) fit to ensure that they behave like white noise. The residual time series plot of the residuals is shown in Figure 4.

The Shapiro–Wilk test for normality yielded a test statistic of $W=0.98911$ with a p-value of $0.5932$. As the p-value is greater than the commonly used significance level of $0.05$, we cannot reject the null hypothesis that the residuals are normally distributed. This indicates that there is no significant evidence to suggest that the residuals deviate from normality, supporting the assumption of normality in the AR(1) model’s residuals. In addition, we plot the ACF of the residuals, which reveals no significant autocorrelation, indicating that the AR(1) model has adequately captured the serial dependence (see Figure 5).

To formally test the null hypothesis that the residuals are uncorrelated, we apply the Box–Ljung test. The test statistic is $Q=0.98261$ with a p-value of 0.3216, indicating that we cannot reject the null hypothesis. Thus, there is no evidence of remaining autocorrelation in the residuals, confirming the adequacy of the AR(1) fit.

The AR(1) model provides a parsimonious and statistically adequate fit for the Nile dataset, effectively capturing the first-order dependence structure. The residual diagnostics, including plots and the Box–Ljung test, confirm that the residuals resemble white noise. This suggests that the AR(1) structure is appropriate for further modeling and simulation efforts related to weighted extropy estimation under dependent data.

To evaluate and compare the performance of the proposed estimator of the weighted extropy under the $\alpha$-mixing dependence condition, we use the Epanechnikov kernel, as it gives the minimum $MSE$ for both the estimators in the simulation. These estimators are then compared with the theoretical value of the weighted extropy in Equation (6) computed from the fitted Gaussian AR(1) model.

Based on the estimated AR(1) coefficient ${\widehat{\varphi }}_1=0.506$ and the innovation variance ${\widehat{\sigma }}^2=21124.84$, the stationary variance is computed as $28405.58$. The theoretical weighted extropy, calculated using the squared stationary Gaussian density, is approximately $-1.373462e-07$. The proposed recursive estimator of the weighted extropy produces a close value of $-1.192312e-07$, confirming its effectiveness in capturing the underlying dependence structure. In contrast, the non-recursive estimator yields a less accurate value of $-0.0002272865$, which deviates significantly from the theoretical benchmark. This comparison demonstrates the superior performance of the recursive estimator in modeling dependent data, confirming its suitability for practical applications involving time series. Consequently, the estimator ${\widehat{\mathcal{J}}}_m^w\left(X\right)$ provides a more accurate approximation of the weighted extropy than ${\widehat{J}}_m^{\ast w}\left(X\right)$ for this dataset. The results from both applications indicate that the proposed estimator is effective and reliable when applied to real-world data scenarios.

The results from both the simulation studies and the data analysis show that the recursive weighted extropy estimator consistently achieves superior performance compared to its non-recursive counterpart in all scenarios.

5. Conclusions

In this paper, we developed a recursive non-parametric estimator for the weighted extropy measure, specifically designed for data sequences exhibiting $\alpha$-mixing dependence. The estimator, denoted ${\widehat{\mathcal{J}}}_m^w\left(X\right)$, allows for efficient online computation by recursively updating as new data points become available. This recursive structure not only offers significant computational advantages, especially for large or streaming datasets, but also enhances adaptability in dynamic systems.

We rigorously derived and proved the key theoretical properties of the estimator, including its recursive formulation, asymptotic unbiasedness, consistency, and asymptotic normality under mild regularity conditions. Together, these properties ensure that the estimator remains statistically reliable and computationally tractable even in the presence of time dependence. Furthermore, simulation studies and data analysis confirm its superior performance in finite samples compared to classical non-recursive kernel-based methods, especially in terms of lower bias and $MSE$.

Beyond the scope of pure statistical theory, the implications of this work extend to several interdisciplinary fields. For example, in reliability engineering, real-time monitoring of systems with dependent failure times can benefit from our recursive approach. In information theory and signal processing, where measurements are often correlated in time, our method supports robust uncertainty quantification.

Looking forward, there are several avenues for further research. One promising direction is to extend the estimation procedure to alternative dependence structures, such as $\rho$-mixing or $\varphi$-mixing, which may better model dependence in certain real-world systems. Another extension involves the recursive estimation of other variants of extropy measures, such as Tsallis extropy, Rényi extropy, etc., especially in reliability or survival settings. Furthermore, exploring multivariate versions of weighted extropy and adapting the recursive estimation framework accordingly would open the door to high-dimensional applications.

In conclusion, this paper makes a rigorous and practically useful contribution to the literature on non-parametric information measures under dependence. The recursive estimator not only strengthens theoretical understanding but also provides a powerful and flexible tool for real-world applications involving dependent data.