Resource-intensive mobile edge computing (MEC) services are often provisioned on a per-request basis, resulting in repeated activation of equivalent service instances and redundant transmission of the same category-level state over overlapping inter-station links. Existing approaches rarely integrate demand aggregation, shared-instance provisioning, and reusable multi-target state distribution into a unified orchestration workflow. This paper proposes Coalition-based Shared Instance Provisioning and Routing (CoSIPR), a shared-service orchestration framework built around dynamic interest coalitions. CoSIPR predicts user requests and mobility, projects predicted locations onto the road network, filters unreliable or infeasible requests, and groups nearby users requesting the same service category. For each coalition, a marginal-gain-based candidate-reduction method and variable neighborhood search determine the serving stations, user assignments, and shared-instance counts. The selected stations then form the target set for a load-aware routing procedure that selects an existing state source and uses path-fusion reinforcement learning (PF-RL) to construct routes that reuse path segments across multiple targets. Experiments using real-world mobility and road-network data show that CoSIPR improves service-category matching, request satisfaction, and the average number of accepted requests per instance. It also reduces aggregate state-transfer cost and limits hotspot exposure while maintaining a controlled trade-off between end-to-end delay and state-transfer cost. These results demonstrate that dynamic interest coalitions and reusable multi-target paths can improve the efficiency of shared-service orchestration in MEC.
In distributed optimization, communication bottlenecks and computational heterogeneity are two critical challenges that undermine system scalability and efficiency. To address these issues, we propose a synchronous algorithm, CC-GT (Conjugate Compressed Gradient Tracking), which integrates conjugate gradient acceleration with dual error-compensated compression to reduce communication overhead and accelerate convergence. Theoretically, we establish its linear convergence under strongly convex and smooth assumptions, and extensive experiments on regression, classification, and neural-network tasks validate the fast convergence and communication efficiency of the synchronous CC-GT. Building on this foundation, we develop GFLCC, an asynchronous extension that eliminates global synchronization barriers while preserving the core mechanisms of CC-GT, thereby inherently mitigating the straggler effect and enabling dynamic load balancing. For the asynchronous GFLCC, extensive comparisons against state-of-the-art asynchronous baselines under harsh network interference demonstrate its strong empirical convergence speed, communication efficiency, and robustness. These results confirm that the proposed framework offers a practical and theoretically grounded solution for communication-constrained and heterogeneous distributed optimization systems.
With the continuous improvement in the high-speed railway (HSR) network, passengers’ holiday travel demand and the scale of tourism passenger flow have grown rapidly, placing higher requirements on HSR transportation services. Current train line plans are formulated according to regular daily passenger flow and fail to adapt to the travel demand generated by various holiday tourism products, resulting in poor adaptation to holiday passenger flow characteristics. Different from existing line-planning studies that consider only regular daily passenger flow, this paper is among the first to embed hierarchical holiday tourism products as a structural input of the HSR line planning problem and to explicitly handle asymmetric passenger demand in holiday periods. To address this gap, this paper optimizes HSR line plans based on the travel characteristics of tourism products. An integer programming model is established that allocates asymmetric passenger flows to train flows while minimizing the total operating cost. The model incorporates both conventional HSR passenger demand and the differentiated travel demand corresponding to different tourism products. A real-world experiment based on the HSR network in Jiangxi Province, China demonstrates that the proposed model effectively improves the tourism transportation efficiency and passenger flow distribution of HSR with reliable practicability. Sensitivity analyses further reveal how the tourism demand scale, the tourism sub-product diversity, and the benchmark line requirements affect the feasibility and cost of the line plan. This study provides a valid reference for the optimization of holiday train line plans and the coordinated development of tourism and transportation.
The rapid adoption of Android applications in mobile commerce has increased exposure to sophisticated malware capable of bypassing traditional security mechanisms through code obfuscation, dynamic code loading, and runtime-triggered malicious behaviors. Although static analysis offers efficient large-scale detection, it often fails to identify concealed runtime activities, while dynamic analysis provides richer behavioral evidence but suffers from limited execution coverage and high computational overhead. To address these complementary limitations, this paper proposes FusionDroid, a lightweight multimodal Android malware detection framework that integrates permission-based static features, permission co-occurrence graph representations, and engineered runtime behavioral features through probability-level ensemble fusion. The framework was evaluated using Android applications collected from the AndroZoo repository, comprising 24,055 valid applications for static analysis and a balanced paired benchmark of 1716 applications for multimodal evaluation. The experimental results show that complementary static and dynamic representations can improve Android malware detection, although the magnitude and nature of the improvement depend on class distribution and evaluation metric. The best-performing model, StackedFusion-LightGBM, achieved 93.31% accuracy, 96.86% precision, 89.53% recall, a 93.05% F1 score, 97.69% ROC–AUC, and 98.14% PR–AUC. A controlled five-fold evaluation on the common paired benchmark further showed an F1 score of 0.9175±0.0176 for the full Static+Graph+Dynamic configuration compared with 0.8781±0.0081 for the Static-only baseline. Paired statistical analysis further supported the improvement. These findings show that multimodal fusion can improve Android malware detection while preserving low-complexity manifest-derived representations. The results support FusionDroid as a staged, sandbox-assisted framework in which lightweight static analysis is complemented by runtime behavioral evidence when deeper inspection is required.
Inverse inclusion problems arise in several applications such as electrical impedance tomography, nondestructive testing, and geophysical imaging. The purpose of this review is to present some recent advances concerning stability estimates for the determination of inclusions from boundary measurements. We focus on elliptic conductivity equations with discontinuous coefficients and discuss different settings including isotropic and anisotropic conductivities, variable coefficients, local boundary measurements, and layered media. Particular attention is devoted to the analytical tools underlying logarithmic stability estimates, namely singular solutions, asymptotic analysis of fundamental solutions, quantitative unique continuation, and propagation of smallness arguments. We also compare different stability regimes, emphasizing how finite-dimensional geometric priors, such as polygonal or polyhedral inclusions, may lead to Hölder or Lipschitz stability estimates in contrast with the logarithmic stability typical of broader infinite-dimensional classes. We highlight how a common methodological framework can be adapted to increasingly complex geometrical and physical configurations. Finally, we briefly discuss some open problems and possible future developments in the study of inverse inclusion problems.
As key components of the niche, ecological factors serve as critical parameters for characterizing the dynamic changes in the quality of ecological environments and determining the equilibrium state of ecosystems. Organisms within an ecosystem generally exhibit an intrinsic tendency to evolve toward an optimal niche; this evolutionary process manifests as higher-order nonlinear characteristics and is accompanied by the inherent uncertainty of ecological factor parameters. Interval Type-2 indirect T-S fuzzy control demonstrates excellent robust stability for a class of higher-order nonlinear systems with parameter uncertainty. This paper introduces the niche proximity function into the consequent of the interval Type-2 indirect T-S fuzzy controller to construct a fuzzy control scheme that integrates biological evolutionary characteristics; it further conducts a theoretical analysis of system stability and convergence, deriving adaptive update laws corresponding to ecological factors. Comparative results from simulation examples indicate that interval Type-2 fuzzy control outperforms Type-1 fuzzy control in terms of both system stability and convergence performance. This conclusion not only reflects the adaptive and self-evolving characteristics of biological organisms and their ability to utilize environmental resources but also validates the excellent intelligent control performance of the proposed method.
The gradual developments in mathematical analysis have increased the demand for improving the efficiency of constraints and their validation, which have a significant contribution to resolving many real-world problems. There are many techniques that are used to modify the fractional inequalities, but all approaches are analytical. The adoption of Machine learning (ML) models is one practical approach to optimize the bounds of inequalities and their numerical validations. In this paper, we implement ML models to modify the bounds of Hermite–Hadamard-type inequalities, which consider the h-Godunova–Levin as a weight function. The classical development of inequalities by generalized fractional operators was never systematically studied to determine which value of the weight gives the best bound possible. We prove a sharp lower bound for the Hermite–Hadamard gap uniform over all admissible weights, show that it is achieved exactly by the classical convex weight, and characterize the optimal weights. To complement this analytical result, a parameterization of the weight function by a feedforward neural network is introduced, which is assumed to be admissible in the h-Godunova–Levin framework, and the validity of the resulting inequalities is proved. A Lipschitz-type error analysis is used to relate the approximation accuracy of the network to the tightness of the bound, and it is shown in numerical experiments that the network learns the optimal weight without any knowledge of the analytical weight’s form. The framework thus ensures the rigor of fractional convexity theory while offering a data-driven method of determining the optimal weight functions in cases where they are not explicitly known. The inherent symmetry properties of the fractional operators and the symmetric structure of the Hermite-Hadamard inequalities are preserved, while the neural network framework introduces a symmetry-breaking mechanism that enables the discovery of optimal weights. This dual perspective on symmetry—both preserving and breaking—provides a comprehensive understanding of the underlying mathematical structures and aligns perfectly with the scope of the journal Symmetry. This practical approach opens a new horizon for researchers and yields better results in the field of analysis.
This paper presents the Adaptive Confidence-Gated Ensemble (ACGE) framework for Network Intrusion Detection Systems (NIDSs) in resource-heterogeneous fog-IoT deployments. No single algorithmic paradigm simultaneously achieves high minority-class detection accuracy, sub-millisecond inference latency, and generalization across heterogeneous IoT traffic at resource-constrained fog-IoT nodes. To address this challenge, ACGE integrates a Sequence Transformer Encoder with LightGBM via a learned and sample-adaptive gating network trained on inter-model error complementarity, formally specified within a three-mode tiered deployment architecture. Both models execute in parallel for every sample; the gating network assigns transformer-dominant weight (α ≈ 1) to 18% of samples where attention-based processing yields the greatest architectural advantage. Under true parallel inference, ACGE latency is 1.45 ms (bounded by the transformer); the measured wall-clock latency under CPU-GPU pipeline parallelism is 1.52 ms. The framework is evaluated under 5-fold stratified cross-validation with McNemar’s significance testing on IoTID20 (primary IoT benchmark) and NSL-KDD (reproducibility anchor) and validated through zero-shot transfer on CICIoT2023 and Edge-IIoTset. Notably, the Sequence Transformer Encoder and ACGE process L = 5 consecutive flow records per decision, providing five times the information available to single-flow baseline models; approximately 57% of the transformer’s performance advantage is attributable to architectural design rather than extended input access. On IoTID20 multiclass classification, ACGE is the only evaluated configuration yielding a negative accuracy-to-macro-F1 gap (−0.51 pp), reaching 99.12 ± 0.03% accuracy and 98.67 ± 0.08% balanced accuracy, demonstrating genuine minority-class improvement. LightGBM achieves the optimal fog-tier profile (0.97 ms latency, concept drift sensitivity of −0.53 pp over three months). On Edge-IIoTset zero-shot transfer, ACGE achieves 93.89% binary accuracy and 85.78% multiclass macro-F1. Ransomware detection (12.34–66.34% F1 across all models) is identified as the primary capability gap; targeted data collection, few-shot adaptation, and federated learning are recommended as the most critical future directions. All inter-paradigm performance differences are reported as the mean ± SD and verified by McNemar’s test with Bonferroni correction.
High penetrations of wind and photovoltaic generation create simultaneous challenges for battery energy storage system (BESS) planning in distribution networks, including differences in nodal regulation value, power-energy configuration, and day-ahead operation under forecast uncertainty. This study develops a sequential planning-to-operation workflow comprising candidate-bus generation, siting and sizing within the candidate set, and finite-scenario day-ahead scheduling for a fixed configuration. First, nodal net-injection sensitivities, Jacobian-assisted pre-screening, and deterministic topology/support safeguards are used to generate the main candidate set, and alternating-current (AC) finite-difference refinement is performed only for the sensitivity-led fast set; in the IEEE-33 system, this refinement reduces the number of AC power-flow calls from 65 for full-node analysis to 17. Next, Sensitivity-Guided Envelope-Based Nonanticipative Adjustable Recourse Optimal Power Flow (SG-ENAR-OPF) is solved separately for each bus in the main candidate set; the BESS location and power/energy capacities are jointly determined subject to the P-Q LinDistFlow model, BESS duration constraints, a shared affine response, and finite-scenario constraints. The full-node audit serves only as an independent paper-level validation benchmark and is not part of the deployable workflow. After the configuration is fixed, interval forecasts for load, photovoltaic (PV) output, and wind-turbine (WT) output at q05/q50/q95 are used to construct 25 static load-renewable disturbance points and seven temporal stress paths, over which a shared finite-scenario day-ahead policy is optimized. The IEEE-33 MAIN case selects Bus 30, with BESS capacities of approximately 10.66 MW/10.66 MWh. Using scaled public time-series data, the final policy is replayed over 46 consecutive 24 h execution windows, comprising 1104 h actual trajectories; under the 0.002 MW/MWh storage-engineering criterion, all 46/46 windows pass storage engineering validation, and energy continuity is maintained across all 45/45 interday boundaries. Further nonlinear AC post-validation converges at all 1104/1104 operating points, of which 1058/1104 satisfy the complete voltage and branch-capacity constraints. Supplementary results for IEEE-69 show that the workflow can be executed on a second radial test feeder. However, the conclusions are strictly limited to the tested feeders, finite scenarios, scaled public-data settings, and stated engineering tolerances and do not constitute a formal guarantee over a continuous uncertainty domain or of general cross-system applicability.
In this study, a new analytical approximation for the Bessel function of the first kind of order two J2(t), t>0, is produced using its power series and its asymptotic series as t tends to infinity. The absolute error between the new analytical approximation and the function J2(t) is analyzed for t∈[0,104], and the results indicate that it decreases significantly as t increases for large values of t. Additionally, our new approximation formula presents estimations for the positive zeros of the function J2(t) with extremely minor relative errors; the largest relative error for the first positive zero is 3.83594×10−5 and, thereafter, the relative errors gradually decrease until they reach 9.91627×10−11 for the twelfth zero. We also showed that our findings outperform some recently published ones.
This study discusses the (2+1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona– Mahony equation, which emerges in weakly nonlinear dispersive plasma waves and shallow water dynamics in ocean engineering. Logarithmic dependent-variable transformations are applied to reconstruct a one-exponential tau function as a common one-soliton profile and derive its dispersion relation. This is a standard transformed solution listed as three normalized logarithmic maps, but not a new family of solutions. Only a one-exponential soliton is claimed, no two-soliton family and no arbitrary-N soliton family. The explicit rational rogue-wave families of the first, second, and third orders are derived using a modified version of a well-known center-shifted polynomial tau-function method that is applied to the KPBBM bilinear form, with both center parameters β and γ independent. The novelty is thus limited to the specific model and is not based on a new KPBBM equation or a fundamentally novel symbolic algorithm. The rogue-wave center translates in the longitudinal and transverse directions through β and γ, respectively, for a fixed order N and fixed model parameters. They leave the pattern, localization width, background, and the arrangement of inner patterns unchanged. Lump solutions and lump–soliton interaction structures are also obtained and investigated. The auxiliary Hirota bilinear constraint and its reduced bilinear representation are explicitly given. The higher-degree equations found in the directional logarithmic maps are not new multilinear equations, but rather the denominator-cleared differential polynomial residuals. The validity of each solution family retained is guaranteed by means of analytical substitution or vanishing of symbolically identical-to-zero residual in the original KP–BBM equation. The two- and three-dimensional plots are used only to demonstrate the amplitude profile, localization, and propagation of the solutions, as verified by the analysis. In the weakly nonlinear, long-wave and weakly transverse regime where the KPBBM reduction is valid, these solutions give idealized mathematical representations of localization and interaction mechanisms. They are not predictive of coastal instability or offshore hydrodynamic loading, for which dimensional calibration and experimental/field validation would be necessary.
Short-term traffic-flow forecasting predicts road-network states, yet spatio-temporal predictors can leave structured, context-dependent residuals. They are seldom reused at inference, while similarity-based transfer can introduce corrections that increase error. We propose Memory-Based Utility-Aligned Residual Evidence with Calibrated Activation for Selective Correction of Traffic-Flow Forecasts (MURECAST), which treats historical residuals as candidate interventions. After freezing a context-aware base forecaster, MURECAST builds a static out-of-sample residual memory from an independent period. At inference, same-node and time-valid constraints delimit records, a forecast-visible utility estimator re-ranks them, and utility-weighted top-K aggregation forms a multi-horizon proposal. A chronological calibration split provides an empirical one-sided lower score for applying the proposal or retaining the base forecast. Across PeMS03, PeMS04, PeMS07, and PeMS08, MURECAST ranked first in 11 of 12 reported dataset–metric comparisons, attaining the lowest mean absolute error (MAE) and root mean squared error (RMSE) on all four datasets and the lowest mean absolute percentage error (MAPE) on three; relative error reductions over the strongest published results were 1.69–9.76%. Non-beneficial corrections represented 17–26% of accepted proposals versus 41–48% of all valid proposals. These results show that MURECAST reuses observed errors while concentrating intervention on corrections with lower observed non-beneficial risk under the evaluated chronological protocol.
Lattice strain engineering, rooted in symmetry-breaking lattice distortion, is an effective strategy for modulating the electronic structure and catalytic performance of electrocatalysts. Herein, non-noble metal CuCoNiCrMox high-entropy carbides with tunable Mo content (HECMo-x) were rapidly synthesized within seconds via a high-temperature shock method. By leveraging composition-dependent lattice distortion engineering to deliberately break local translational symmetry, these catalysts were developed to optimize the alkaline hydrogen evolution reaction (HER). Density functional theory calculations reveal that lattice distortion optimizes the d-band center and regulates the electronic configuration. Concurrently, kinetic isotope effect tests and variable-potential electrochemical impedance spectroscopy measurements verify that this modulation balances the reaction kinetics of water dissociation and hydrogen adsorption, thereby accelerating the alkaline HER process. Consequently, the optimized HECMo-15% electrocatalyst exhibits outstanding activity, requiring an overpotential of only 34 mV at 10 mA cm−2. Furthermore, it exposes abundant active sites and maintains long-term operational stability with negligible attenuation over 23 h. This work provides a feasible design strategy and a practical paradigm for developing non-noble metal high-entropy carbides as highly efficient electrocatalysts for energy conversion applications.
This review examined cumulative findings from recent meta-analyses to identify current challenges and possible suggestions for improving the effects of BCI systems on stroke motor recovery. Consistent with the Preferred Reporting Items for Systematic Reviews and Meta-Analyses guidelines, a systematic literature search was conducted using PubMed, Web of Science, and the Cochrane Library on 30 June 2026. A total of 17 systematic reviews and meta-analyses were included. Among three motor intent-induced (i.e., motor attempt, motor observation, and motor imagery) modalities, motor attempt was the modality most consistently associated with significant therapeutic effects across meta-analyses. Electrical stimulation was a consistently effective feedback modality, whereas robot-assisted and visual feedback showed heterogeneous effects. Higher weekly session frequencies and moderate session durations (approximately 20–60 min) showed consistent motor recovery. Stroke type, age, intervention period, total sessions, total training time, and long-term effect durability were inconsistent across the included evidence. These findings suggest that applying BCI-based training is an effective rehabilitation program for the functional recovery of upper extremities in patients with stroke who have moderate to severe motor impairments, potentially achieving greater therapeutic efficacy when combining motor attempts with electrical stimulation.
Multimodal aspect-based sentiment analysis (MABSA) predicts the sentiment expressed toward a target aspect by jointly using textual and visual information, supporting fine-grained opinion analysis in product reviews, brand monitoring, and customer feedback. However, existing approaches remain sensitive to irrelevant visual regions, weak text–image alignment, and limited use of external knowledge. Motivated by these challenges, this study systematically evaluated two text-only large language models and four open-weight large vision-language models for aspect-level sentiment classification. The open-weight models were adapted using 4-bit quantized low-rank adaptation, while GPT-4o was assessed under zero-shot, one-shot, and five-shot in-context learning without parameter updates. Experiments were conducted on Twitter-2015, Twitter-2017, and the seven-domain MASAD dataset and evaluated using accuracy and macro-F1. Among the evaluated multimodal models, Qwen3-VL-8B-Instruct achieves the strongest performance, reaching 83.22% accuracy and 81.72% macro-F1 on Twitter-2015, 79.50% and 78.93% on Twitter-2017, and up to 99.84% and 99.83% in the Plant domain of MASAD. From a symmetry perspective, semantically aligned text–image–aspect inputs provide consistent cross-modal evidence, whereas shuffled images introduce asymmetric, symmetry-breaking information. The resulting performance degradation under shuffled-image ablation indicates that reliable aspect-level sentiment prediction depends on preserving cross-modal semantic correspondence. These findings demonstrate the effectiveness of parameter-efficient LVLM adaptation for MABSA.
In this work, we study the relationship among Lie algebraic homotopy 3-types. We investigate the categorical connection between crossed squares, internal crossed modules, 2-crossed modules, quadratic modules of Lie algebras, and simplicial and bisimplicial Lie algebras.
Epilepsy EEG signal recognition is an important method for epilepsy detection. However, the traditional machine learning methods used for epilepsy EEG signal recognition face many challenges. For example, traditional intelligent recognition methods often ignore the imbalance of epilepsy EEG signals, which misclassifies positive samples and may cause serious consequences; the existing imbalanced classification methods ignore class-guided global inter-class correlations among samples, resulting in poor classification results. In order to overcome these challenges, a cost-sensitive-based TSK method (CITSK) is proposed for imbalanced epilepsy EEG signal recognition. This proposed method constructs a Class-guided Sample Collaborative Relationship (SCR) matrix to assign higher weight to positive minority samples, and supplements global inter-class collaborative penalty information on the basis of the traditional static cost-sensitive matrix. Then, a model combining the SCR matrix and TSK is proposed, which inherits the good generalization ability and interpretability of TSK while ensuring classification performance. Experiments on real EEG datasets demonstrate the effectiveness of the proposed method and its applicability to the imbalanced classification of epileptic EEG signals. Quantitatively, the CITSK method achieves an average Accuracy of 93.96%, an average F_measure of 88.33%, and an average G_means of 91.34% across various imbalanced datasets. Furthermore, comparative evaluations demonstrate that the proposed CITSK achieves superior overall performance across comprehensive evaluation metrics compared to baseline models, offering a highly robust and interpretable solution for imbalanced data scenarios via reasonable global inter-class penalty compensation.
Spacecraft radiation assessment requires expert interpretation of continuous environment-model outputs. We present a reflection-equivariant Mamdani fuzzy system for relative triage of modelled total ionising dose (TID) and solar-proton exposure. Radiation environment severity and solar-proton severity are derived from OMERE 5.9.5 runs of the AE9/AP9 (IRENE 1.57.004, mean mode) and Emission of Solar Protons (ESP, 90 per cent confidence) models, and, together with mission duration, are mapped through reflection-paired membership partitions and a 27-rule sum-based rule base to four triage categories. We prove reflection symmetry of the input and output partitions, permutation symmetry of the rule map, risk-reversal duality of the aggregated inference and centroid score, and reflection equivariance of a normalised output-support vector retained before defuzzification. The architecture is examined on nine reference mission scenarios and additional boundary cases using sensitivity, comparative-variant and cumulative-versus-duration-normalised analyses. The results show exact algebraic consistency with the imposed symmetry identities and transparent rule-level traceability, while also revealing the small local non-monotonicity of the centroid score and formulation sensitivity in the seven-year GLONASS-like scenario. Under the integrated-exposure formulation, scores range from 0.381 for the polar low-Earth-orbit scenario to 0.892 for the geostationary orbit (GEO). Because the same nine scenarios also define the frozen normalisation anchors, this range is a reference-set demonstration rather than an out-of-sample result. Evaluation to date comprises internal mathematical-consistency checks, comparison with an author-defined conservative heuristic and concordance with a seven-member expert panel blinded to the model output but rating the same scenario descriptions; the system has not been validated against ground-truth radiation-hardness outcomes such as mission anomaly records or component-qualification results. Cases for which the integrated and duration-normalised diagnostics disagree are flagged for separate engineering analysis. The system is a reference-benchmarked proof-of-concept pre-screening method and does not replace project-specific TID, total non-ionising dose (TNID), single-event-effect, shielding or component-qualification analysis.
We establish an asymptotic theory for the Jones inverse-weighted kernel density estimator when length-biased observations form a strictly stationary short-range dependent sequence. The statistical difficulty is intrinsically composite: reciprocal weighting is singular at the origin, the normalizing mean is estimated from the same dependent sample, kernel localization shrinks with the bandwidth, and the centered summands form a row-wise stationary triangular array whose envelope diverges at rate hn−1. Under a non-negative compactly supported Lipschitz kernel, an inverse-moment condition, geometric α-mixing, local regularity of the target density, and uniform local bounds on lagged bivariate densities, we prove strong uniform consistency on compact subsets of (0,∞) and, separately, the uniform stochastic bound OP{hn2+(logn/(nhn))1/2}. A covariance-localization argument shows that the scaled serial-covariance contribution is O{hnlog(1/hn)}=o(1), so the first-order pointwise variance coincides with that of the corresponding independent length-biased estimator. Pointwise and finite-dimensional Gaussian limits are obtained by an explicit big-block/small-block argument with off-diagonal covariance control. The ratio normalization is treated directly: its variance contribution, its product with the localized fluctuation, and its cross-covariance with that fluctuation are all negligible at the nhn scale. We further derive first-order AMSE and AMISE criteria, their oracle bandwidths, and feasible pointwise studentization under undersmoothing. The numerical study separates oracle from data-driven bandwidth selection, evaluates full-ratio HAC and moving-block corrections, examines a Frank-copula Markov robustness design, and benchmarks the Jones estimator against an alternative length-biased estimator. The simulations support the first-order theory while demonstrating that persistent short-range dependence can remain consequential for finite-sample uncertainty.
The Weibull distribution is one of the most versatile and widely applied continuous probability distributions in reliability engineering, survival analysis, wind-energy modelling, and extreme-value theory. Classical parameter estimation relies on maximum likelihood estimation or method-of-moments using integer-order moments, both of which may suffer from instability or high variance in small to moderate samples. This paper develops a rigorous framework for estimating Weibull parameters by combining the maximum-entropy principle with fractional-order moment constraints, i.e., constraints of the form E[Xr] for non-integer r>0. A central theme of the paper is symmetry: we show that the maximum-entropy density subject to a finite set of fractional-moment constraints uniquely recovers the Weibull family, and that the underlying moment-matching system, while not symmetric in every sense considered in an earlier draft (see Remark 3), possesses a precisely characterised duality under joint rescaling and relabelling of the exponent pair, together with a log-moment map whose local curvature is strictly positive and increasing with exponent spacing rather than symmetric about a fixed midpoint. We derive closed-form expressions relating the Lagrange multipliers to the shape and scale parameters, establish new sound theoretical results on the symmetric behaviour of the moment-ratio function and its sensitivity, and propose a numerically stable algorithm for solving the resulting moment-matching system. Extensive Monte Carlo experiments demonstrate that the proposed maximum-entropy fractional-moment estimator achieves a bias and root-mean-square error that are comparable to, and for small-to-moderate samples somewhat better than, maximum likelihood estimation, with the size of the advantage depending on how closely the chosen exponent pair tracks the true shape parameter. Applications to real wind-speed data and composite-material fatigue life data illustrate the practical utility of the method. The paper contributes both to the information-theoretic foundations of distribution fitting and to applied statistical methodology, with symmetry serving as both a diagnostic tool and a unifying structural principle throughout.
