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In this paper, estimation for the modified Lindley distribution parameter is studied based on progressive Type II censored data. Maximum likelihood estimation, Pivotal estimation, and Bayesian estimation were calculated using the Lindley approximation and Markov chain Monte Carlo methods. Asymptotic, Pivotal, bootstrap, and Bayesian confidence intervals are provided. A Monte Carlo simulation study has been conducted to evaluate and compare the performance of different estimation methods. To further illustrate the introduced estimation methods, two real examples are provided.

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Studying various models in queueing theory is essential for improving the efficiency of queueing systems. In this paper, from the family of models {E_r/M/c; r,c in N}, the E_r/M/3 model is introduced, and quantities such as the distribution of the number of customers in the system, the average number of customers in the queue and in the system, and the average waiting time in the queue and in the system for a single customer are obtained. Given the crucial role of the traffic intensity parameter in performance evaluation criteria of queueing systems, this parameter is estimated using Bayesian, E‑Bayesian, and hierarchical Bayesian methods under the general entropy loss function and based on the system’s stopping time. Furthermore, based on the E‑Bayesian estimator, a new estimator for the traffic intensity parameter is proposed, referred to in this paper as the E^2‑Bayesian estimator. Accordingly, among the Bayesian, E‑Bayesian, hierarchical Bayesian, and the new estimator, the one that minimizes the average waiting time in the customer queue is considered the optimal estimator for the traffic intensity parameter in this paper. Finally, through Monte Carlo simulation and using a real dataset, the superiority of the proposed estimator over the other mentioned estimators is demonstrated.

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This study investigates the wavelet energy distribution in high-frequency fractal systems and analyzes its characteristics using information-theoretic measures. The main innovation of this paper lies in modeling the wavelet energy distribution ($p_j$) using a truncated geometric distribution and incorporating the concept of extropy to quantify system complexity. It is demonstrated that this distribution is strongly influenced by the fractal parameter $alpha$ and the number of decomposition levels $M$. By computing wavelet entropy and extropy as measures of disorder and information, respectively—the study provides a quantitative analysis of the complexity of these systems. The paper further examines key properties of this distribution, including its convergence to geometric, uniform, and degenerate distributions under limiting conditions (e.g., $M to infty$ or $alpha to 0$). Results indicate that entropy and extropy serve as complementary tools for a comprehensive description of system behavior: while entropy measures disorder, extropy reflects the degree of information and certainty. This approach establishes a novel framework for analyzing real-world signals with varying parameters and holds potential applications in the analysis of fractal signals and modeling of complex systems in fields such as finance and biology.

To validate the theoretical findings, synthetic fractal signals (fractional Brownian motion) with varying fractal parameters ($alpha$) and decomposition levels ($M$) were simulated. Numerical results show that wavelet entropy increases significantly with the number of decomposition levels ($M$), whereas extropy exhibits slower growth and saturates at higher decomposition levels. These findings underscore the importance of selecting an appropriate decomposition level. The proposed combined framework offers a powerful tool for analyzing and modeling complex, non-stationary systems in domains such as finance and biology.