Estimation of Weighted Extropy Under the α-Mixing Dependence Condition

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 α-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.