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31 pages, 3479 KB  
Article
MV-S2CD: A Modality-Bridged Vision Foundation Model-Based Framework for Unsupervised Optical–SAR Change Detection
by Yongqi Shi, Ruopeng Yang, Changsheng Yin, Yiwei Lu, Bo Huang, Yongqi Wen, Yihao Zhong and Zhaoyang Gu
Remote Sens. 2026, 18(6), 931; https://doi.org/10.3390/rs18060931 - 19 Mar 2026
Viewed by 358
Abstract
Unsupervised change detection (UCD) from heterogeneous bitemporal optical–SAR imagery is challenging due to modality discrepancy, speckle/illumination variations, and the absence of change annotations. We propose MV-S2CD, a vision foundation model (VFM)-based framework that learns a modality-bridged latent space and produces dense change maps [...] Read more.
Unsupervised change detection (UCD) from heterogeneous bitemporal optical–SAR imagery is challenging due to modality discrepancy, speckle/illumination variations, and the absence of change annotations. We propose MV-S2CD, a vision foundation model (VFM)-based framework that learns a modality-bridged latent space and produces dense change maps in a fully unsupervised manner. To robustly adapt pretrained VFM priors to heterogeneous inputs with minimal task-specific parameters, MV-S2CD incorporates lightweight modality-specific adapters and parameter-efficient low-rank adaptation (LoRA) in high-level layers. A shared projector embeds the two observations into a common geometry, enabling consistent cross-modal comparison and reducing sensor-induced domain shift. Building on the bridged representation, we design a dual-branch change reasoning module that decouples structure-sensitive cues from semantic-consistency cues: a structure pathway preserves fine boundaries and local variations, while a semantic-consistency pathway employs reliability gating and multi-scale context aggregation to suppress pseudo-changes caused by modality-specific nuisances and residual misregistration. For label-free optimization, we develop a difference-centric self-supervision scheme with two perturbation views and reliability-guided pseudo-partitioning, jointly enforcing pseudo-unchanged invariance, pseudo-changed/unchanged separability, and sparsity and edge-preserving regularization. Experiments on three heterogeneous optical–SAR benchmarks demonstrate that MV-S2CD consistently improves the Precision–Recall trade-off and achieves state-of-the-art performance among unsupervised baselines, while remaining backbone-flexible and efficient. Full article
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9 pages, 2524 KB  
Proceeding Paper
Wide-Area GNSS Interference Source Localization Using a Sparse Monitoring Network
by Aiden Morrison and Nadezda Sokolova
Eng. Proc. 2026, 126(1), 29; https://doi.org/10.3390/engproc2026126029 - 25 Feb 2026
Viewed by 399
Abstract
This paper discusses the design, development, and initial testing of a distributed monitoring system intended to detect and localize sources of harmful interference impacting Global Navigation Satellite System (GNSS) users over city-sized areas using only a small number of monitoring stations to limit [...] Read more.
This paper discusses the design, development, and initial testing of a distributed monitoring system intended to detect and localize sources of harmful interference impacting Global Navigation Satellite System (GNSS) users over city-sized areas using only a small number of monitoring stations to limit costs. The motivation and background of the work is rooted in the results of the Advanced Radio Frequency Interference Detection Analysis and Alerting System (ARFIDAAS), a network of GNSS Radio Frequency Interference (RFI) monitors which built the largest known database of multi-frequency GNSS RFI events. Insights gained from this database on parameters such as modulations, impacted bands, power-level distributions and other relevant factors are used to inform the design of the source localization system discussed in the paper. The design of the receiver hardware to allow the implementation of a distributed Time Difference of Arrival (TDOA) detection and localization system incorporating components of Commercial Off-The-Shelf (COTS) radios while supporting dynamic coverage of all L-band signals is detailed, along with the software architecture used to control and operate the individual nodes of the work-in-progress development systems and testbed. Further information is included to describe the design and operation of the software which controls the composite network, including decisions made for the support of mobile detectors and multiple data consumers to allow the pursuit of multiple simultaneous sources. Since the system is designed for the detection of sources which are likely below the local noise floor at the participating nodes, the paper explores the derived operating envelope of the architecture, showing examples of measurements produced during controlled field testing at Jammertest 2023, and discusses considerations for the screening of nuisance events that are likely to be unintentionally generated by incidental devices over a city-sized area. Full article
(This article belongs to the Proceedings of European Navigation Conference 2025)
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19 pages, 486 KB  
Article
Late-Time Constraints on Future Singularity Dark Energy Models from Geometry and Growth
by Tomasz Denkiewicz
Universe 2026, 12(1), 14; https://doi.org/10.3390/universe12010014 - 3 Jan 2026
Viewed by 406
Abstract
We confront two future-singularity dark-energy templates—sudden future singularities (SFSs) and finite scale factor singularities (FSFSs)—with late-time geometric probes and redshift-space distortion growth data. We compute the observable growth fσ8(z) by solving the full linear perturbation system (including the [...] Read more.
We confront two future-singularity dark-energy templates—sudden future singularities (SFSs) and finite scale factor singularities (FSFSs)—with late-time geometric probes and redshift-space distortion growth data. We compute the observable growth fσ8(z) by solving the full linear perturbation system (including the standard fiducial cosmology rescaling of RSD measurements) and build a joint χ2 from Pantheon+SH0ES SNe Ia, H(z), DESI AP-only BAO, and fσ8. Parameter constraints are obtained via grid-based profiling over nuisance parameters and the singularity time location parameter. We compare the viability and goodness of fit of the singularity scenarios to the ΛCDM reference. Full article
(This article belongs to the Section Cosmology)
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17 pages, 655 KB  
Article
Confidence Intervals for Two Proportions—A Generalized Estimation and Information Assessment
by Qiang Wu and Paul Vos
Mathematics 2026, 14(1), 45; https://doi.org/10.3390/math14010045 - 23 Dec 2025
Viewed by 486
Abstract
This paper develops profile score confidence intervals (i.e., z-standard intervals) by inverting orthogonalized score estimators for comparing two independent binomial proportions in rate difference, rate ratio, or odds ratio and utilizes a generalized estimation framework of Vos and Wu (Inf. Geom. 8: [...] Read more.
This paper develops profile score confidence intervals (i.e., z-standard intervals) by inverting orthogonalized score estimators for comparing two independent binomial proportions in rate difference, rate ratio, or odds ratio and utilizes a generalized estimation framework of Vos and Wu (Inf. Geom. 8: 99–123, 2025) to evaluate different confidence interval methods. The orthogonalized score estimators dominate other generalized estimators in Λ-efficiency for distinguishing parameter values around the truth, so the z-standard intervals become more efficient and acquire coverage closer to the nominal level than other types of confidence intervals. In addition, the degrees of freedom and small-sample corrections (applied to the profile nuisance parameter estimates) are expected to improve the coverage of the z-standard intervals and help maintain them above the nominal level. Computational algorithms are developed to find the z-standard intervals using R’s polyroot function. Numerical studies are conducted to compare both coverage and endpoints of different types of confidence intervals. Full article
(This article belongs to the Section D1: Probability and Statistics)
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14 pages, 483 KB  
Article
Odor Impact of Municipal Waste Biogas Plants—Statistical Analysis of Annual Results
by Marta Wiśniewska, Krystyna Lelicińska-Serafin, Andrzej Kulig and Piotr Manczarski
Energies 2026, 19(1), 58; https://doi.org/10.3390/en19010058 - 22 Dec 2025
Viewed by 589
Abstract
The amount of municipal solid waste (MSW) generated worldwide is constantly growing. In many countries, anaerobic digestion (AD) is the recommended process for converting organic waste, playing a crucial role in the transition to a circular economy. Capturing and using biogas helps to [...] Read more.
The amount of municipal solid waste (MSW) generated worldwide is constantly growing. In many countries, anaerobic digestion (AD) is the recommended process for converting organic waste, playing a crucial role in the transition to a circular economy. Capturing and using biogas helps to reduce greenhouse gas emissions. This paper summarizes the results of comprehensive studies conducted at three municipal waste biogas plants (MWBPs) located in Poland. These studies include measurements related to concentrations of odor (cod) and odorants (C) as well as microclimate parameters. We statistically analyzed the research obtained. However, the microclimatic parameters were not used in a final PCA model and were only used in exploratory correlation. Principal component analysis (PCA) is one of the methods of statistical factor analysis, which allows for the organization of a large set of data from three objects from the annual study. The use of PCA allowed us to determine which substance at a specific biogas plant is primarily responsible for odor nuisance and to estimate the percentage of variability contained in the first two principal components. The obtained results clearly indicate the influence of the technological regime and the type of fermentation feed on the determining effect of a specific odorant. In connection with the vision of creating new MWBPs that are consistent with circular economy assumptions, it seems advisable to extend the conducted analysis to include an immission study outside the plant boundaries. This study could play a crucial role in public consultations and serve as a tool for minimizing odor nuisance. Full article
(This article belongs to the Special Issue Biomass, Biofuels and Waste: 3rd Edition)
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21 pages, 991 KB  
Article
Hybrid Cramér-Rao Bound for Quantum Bayes Point Estimation with Nuisance Parameters
by Jianchao Zhang and Jun Suzuki
Entropy 2025, 27(12), 1184; https://doi.org/10.3390/e27121184 - 21 Nov 2025
Viewed by 846
Abstract
We develop a hybrid framework for quantum parameter estimation in the presence of nuisance parameters. In this scheme, the parameters of interest are treated as fixed non-random parameters while nuisance parameters are integrated out with respect to a prior (random parameters). Within this [...] Read more.
We develop a hybrid framework for quantum parameter estimation in the presence of nuisance parameters. In this scheme, the parameters of interest are treated as fixed non-random parameters while nuisance parameters are integrated out with respect to a prior (random parameters). Within this setting, we introduce the hybrid partial quantum Fisher information matrix (hpQFIM), defined by prior-averaging the nuisance block of the QFIM and taking a Schur complement, and derive a corresponding Cramér–Rao-type lower bound on the hybrid risk. We establish the structural properties of the hpQFIM, including inequalities that bracket it between computationally tractable approximations, as well as limiting behaviors under extreme priors. Operationally, the hybrid approach improves over pure point estimation since the optimal measurement for the parameters of interest depends only on the prior distribution of the nuisance, rather than on its unknown value. We illustrate the framework with analytically solvable qubit models and numerical examples, clarifying how partial prior information on nuisance variables can be systematically exploited in quantum metrology. Full article
(This article belongs to the Special Issue Quantum Measurements and Quantum Metrology)
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16 pages, 387 KB  
Article
Estimation and Sufficiency Under the Mixed Effects Extended Growth Curve Model with Compound Symmetry Covariance Structure
by Katarzyna Filipiak, Augustyn Markiewicz, Paweł Krajewski and Hanna Ćwiek-Kupczyńska
Symmetry 2025, 17(11), 1901; https://doi.org/10.3390/sym17111901 - 7 Nov 2025
Viewed by 416
Abstract
An extended growth curve model with fixed and random effects is considered. Under the assumption of multivariate normality, the maximum likelihood estimators of the fixed effects and the dispersion matrix are determined in a model with random nuisance parameters, both without any assumption [...] Read more.
An extended growth curve model with fixed and random effects is considered. Under the assumption of multivariate normality, the maximum likelihood estimators of the fixed effects and the dispersion matrix are determined in a model with random nuisance parameters, both without any assumption on the covariance structure and under the assumption of compound symmetry. For this purpose, rules for differentiation of symmetric matrices are applied. Furthermore, when the experiments are designed in balanced complete blocks, particular symmetric matrices appear in the likelihood equations, allowing closed-form expressions for the estimators. It is also shown that the vector of sufficient statistics for the fixed effects extended growth curve model is also sufficient for the model with random nuisance parameters. The presented results are illustrated using a real data example. Full article
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15 pages, 545 KB  
Article
Geometry of Statistical Manifolds
by Paul W. Vos
Entropy 2025, 27(11), 1110; https://doi.org/10.3390/e27111110 - 27 Oct 2025
Viewed by 1394
Abstract
A statistical manifold M can be defined as a Riemannian manifold each of whose points is a probability distribution on the same support. In fact, statistical manifolds possess a richer geometric structure beyond the Fisher information metric defined on the tangent bundle [...] Read more.
A statistical manifold M can be defined as a Riemannian manifold each of whose points is a probability distribution on the same support. In fact, statistical manifolds possess a richer geometric structure beyond the Fisher information metric defined on the tangent bundle TM. Recognizing that points in M are distributions and not just generic points in a manifold, TM can be extended to a Hilbert bundle HM. This extension proves fundamental when we generalize the classical notion of a point estimate—a single point in M—to a function on M that characterizes the relationship between observed data and each distribution in M. The log likelihood and score functions are important examples of generalized estimators. In terms of a parameterization θ:MΘRk, θ^ is a distribution on Θ while its generalization gθ^=θ^Eθ^ as an estimate is a function over Θ that indicates inconsistency between the model and data. As an estimator, gθ^ is a distribution of functions. Geometric properties of these functions describe statistical properties of gθ^. In particular, the expected slopes of gθ^ are used to define Λ(gθ^), the Λ-information of gθ^. The Fisher information I is an upper bound for the Λ-information: for all g, Λ(g)I. We demonstrate the utility of this geometric perspective using the two-sample problem. Full article
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21 pages, 3469 KB  
Article
Bayesian Discrepancy Measure: Higher-Order and Skewed Approximations
by Elena Bortolato, Francesco Bertolino, Monica Musio and Laura Ventura
Entropy 2025, 27(7), 657; https://doi.org/10.3390/e27070657 - 20 Jun 2025
Viewed by 1340
Abstract
The aim of this paper is to discuss both higher-order asymptotic expansions and skewed approximations for the Bayesian discrepancy measure used in testing precise statistical hypotheses. In particular, we derive results on third-order asymptotic approximations and skewed approximations for univariate posterior distributions, including [...] Read more.
The aim of this paper is to discuss both higher-order asymptotic expansions and skewed approximations for the Bayesian discrepancy measure used in testing precise statistical hypotheses. In particular, we derive results on third-order asymptotic approximations and skewed approximations for univariate posterior distributions, including cases with nuisance parameters, demonstrating improved accuracy in capturing posterior shape with little additional computational cost over simple first-order approximations. For third-order approximations, connections to frequentist inference via matching priors are highlighted. Moreover, the definition of the Bayesian discrepancy measure and the proposed methodology are extended to the multivariate setting, employing tractable skew-normal posterior approximations obtained via derivative matching at the mode. Accurate multivariate approximations for the Bayesian discrepancy measure are then derived by defining credible regions based on an optimal transport map that transforms the skew-normal approximation to a standard multivariate normal distribution. The performance and practical benefits of these higher-order and skewed approximations are illustrated through two examples. Full article
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22 pages, 1651 KB  
Article
Stress and the City: Body Condition, Blood Parameters, Parasite Load, and Stomach Calorimetry of Rural and Urban European Rabbit Populations
by Madlen Fellmeth, Denise Babitsch, Anne Madel, Marie-Luise Schrödl, Marie-Christin Uhde, Angela Schießl, Bruno Streit, Markus Weinhardt and Bernd Hermann
Wild 2025, 2(2), 23; https://doi.org/10.3390/wild2020023 - 16 Jun 2025
Viewed by 1490
Abstract
(1) Background: We combined physiological and morphological data of the European rabbit (Oryctolagus cuniculus) to provide insights into the question of how urbanization affects the health of urban wildlife populations. (2) Methods: We dissected 39 urban and 34 rural wild rabbits [...] Read more.
(1) Background: We combined physiological and morphological data of the European rabbit (Oryctolagus cuniculus) to provide insights into the question of how urbanization affects the health of urban wildlife populations. (2) Methods: We dissected 39 urban and 34 rural wild rabbits in order to compare organ weights, as well as stomach contents. Furthermore, we collected blood and fecal samples. (3) Results: Rural rabbits had a significantly longer body and a higher body weight as well as more fat tissue around their kidneys compared to urban rabbits. In contrast, the stomach, the intestines, the liver, the lung, and the brain of urban rabbits were significantly heavier. The amount of hematocrit, hemoglobin, and the mean corpuscular volume was significantly higher in urban rabbits. The caloric energy content of the stomach was comparable between rural and urban rabbits and was merely influenced by the season being higher in autumn. Rural rabbits had an overall higher mean parasite index compared to urban rabbits. (4) Conclusions: The results of our study allow for a deeper understanding of how density-dependent (e.g., transmission of diseases) and density-independent factors (e.g., food quality) influence the health status and life history traits of urban wildlife populations compared to their rural counterparts. Full article
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9 pages, 453 KB  
Article
Constraints on Lorentz Invariance Violation from Gamma-Ray Burst Rest-Frame Spectral Lags Using Profile Likelihood
by Vyaas Ramakrishnan and Shantanu Desai
Universe 2025, 11(6), 183; https://doi.org/10.3390/universe11060183 - 6 Jun 2025
Cited by 4 | Viewed by 1474
Abstract
We reanalyze the spectral lag data for 56 Gamma-Ray Bursts (GRBs) in the cosmological rest frame to search for Lorentz Invariance Violation (LIV) using frequentist inference. For this purpose, we use the technique of profile likelihood to deal with the nuisance parameters, corresponding [...] Read more.
We reanalyze the spectral lag data for 56 Gamma-Ray Bursts (GRBs) in the cosmological rest frame to search for Lorentz Invariance Violation (LIV) using frequentist inference. For this purpose, we use the technique of profile likelihood to deal with the nuisance parameters, corresponding to a constant time lag in the GRB rest frame and an unknown intrinsic scatter, while the parameter of interest is the energy scale for LIV (EQG). With this method, we do not obtain a global minimum for χ2 as a function of EQG up to the Planck scale. Thus, we can obtain one-sided lower limits on EQG in a seamless manner. Therefore, the 95% c.l. lower limits which we thus obtain on EQG are then given by EQG2.07×1014 GeV and EQG3.71×105 GeV, for linear and quadratic LIV, respectively. Full article
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19 pages, 2508 KB  
Article
A Novel Moving Load Identification Method for Continuous Rigid-Frame Bridges Using a Field-Based Displacement Influence Line
by Linyong Deng, Ping Liu, Tianli Huang and Sakdirat Kaewunruen
Appl. Sci. 2025, 15(11), 6028; https://doi.org/10.3390/app15116028 - 27 May 2025
Cited by 1 | Viewed by 1463
Abstract
This study focuses on a new identification method for moving loads on bridge structures using field-based displacement data from different measurement points on a continuous rigid-frame bridge. A novel approach has been proposed to make use of the area of the absolute field [...] Read more.
This study focuses on a new identification method for moving loads on bridge structures using field-based displacement data from different measurement points on a continuous rigid-frame bridge. A novel approach has been proposed to make use of the area of the absolute field value derived from the displacement influence line of continuous rigid-frame bridges. Considering the potential presence of other nuisance loads (i.e., noise) on the bridge, this method can significantly mitigate the impact of noise by adopting the absolute area method of influence lines. In addition, the new method combines data from various field measurement points to identify the moving loads, which can in turn minimize the influence of measurement errors. To validate the new method, several numerical simulations varying different noises and parameters have been carried out for benchmarking. The results show that our proposed method achieves an outstanding identification accuracy of over 95% for the simulation cases with the disturbance noise amplitude less than 1.0% and the field data with random noise. This new method enables the identification of moving loads on bridges, thereby providing fundamental data for bridge health monitoring and damage detection. This will help improve predictability of the remaining fatigue life of bridge structures. Full article
(This article belongs to the Section Civil Engineering)
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24 pages, 7986 KB  
Article
Employing Eye Trackers to Reduce Nuisance Alarms
by Katherine Herdt, Michael Hildebrandt, Katya LeBlanc and Nathan Lau
Sensors 2025, 25(9), 2635; https://doi.org/10.3390/s25092635 - 22 Apr 2025
Viewed by 1337
Abstract
When process operators anticipate an alarm prior to its annunciation, that alarm loses information value and becomes a nuisance. This study investigated using eye trackers to measure and adjust the salience of alarms with three methods of gaze-based acknowledgement (GBA) of alarms that [...] Read more.
When process operators anticipate an alarm prior to its annunciation, that alarm loses information value and becomes a nuisance. This study investigated using eye trackers to measure and adjust the salience of alarms with three methods of gaze-based acknowledgement (GBA) of alarms that estimate operator anticipation. When these methods detected possible alarm anticipation, the alarm’s audio and visual salience was reduced. A total of 24 engineering students (male = 14, female = 10) aged between 18 and 45 were recruited to predict alarms and control a process parameter in three scenario types (parameter near threshold, trending, or fluctuating). The study evaluated whether behaviors of the monitored parameter affected how frequently the three GBA methods were utilized and whether reducing alarm salience improved control task performance. The results did not show significant task improvement with any GBA methods (F(3,69) = 1.357, p = 0.263, partial η2 = 0.056). However, the scenario type affected which GBA method was more utilized (X2 (2, N = 432) = 30.147, p < 0.001). Alarm prediction hits with gaze-based acknowledgements coincided more frequently than alarm prediction hits without gaze-based acknowledgements (X2 (1, N = 432) = 23.802, p < 0.001, OR = 3.877, 95% CI 2.25–6.68, p < 0.05). Participant ratings indicated an overall preference for the three GBA methods over a standard alarm design (F(3,63) = 3.745, p = 0.015, partial η2 = 0.151). This study provides empirical evidence for the potential of eye tracking in alarm management but highlights the need for additional research to increase validity for inferring alarm anticipation. Full article
(This article belongs to the Special Issue New Trends in Biometric Sensing and Information Processing)
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21 pages, 2376 KB  
Article
Ground-Based Green Façade for Enhanced Greywater Treatment and Sustainable Water Management
by Nisreen Obeidat, Ahmad Abu Awwad, Ahmed Al-Salaymeh, Riccardo Bresciani, Fabio Masi, Anacleto Rizzo, Jomanah AlBtoosh and Mutaz M. Zoubi
Water 2025, 17(3), 346; https://doi.org/10.3390/w17030346 - 26 Jan 2025
Cited by 7 | Viewed by 3233
Abstract
Urban areas face challenges with water scarcity, and the use of non-conventional water resources for uses not requiring potable quality is being promoted more and more by governments and international agencies. However, non-conventional water resources, such as rainwater and greywater, need to be [...] Read more.
Urban areas face challenges with water scarcity, and the use of non-conventional water resources for uses not requiring potable quality is being promoted more and more by governments and international agencies. However, non-conventional water resources, such as rainwater and greywater, need to be treated before use to avoid health risks and possible nuisance (smell, bacteria and algae proliferation in storage tanks, etc.). This study is aimed at demonstrating the feasibility of a system reusing treated greywater for toilet flushing, relying on a nature-based treatment technology—ground-based green façades—with limited maintenance requirements which is therefore easily replicable for decentralized treatment systems, like those of greywater reuse at building scales. The demonstrative system has been installed at the University of Jordan’s Al-Zahra dormitory in Amman and uses a degreaser as the primary treatment followed by ground-based green façade technology as a secondary treatment mechanism. The effluent is stored in an underground tank and directed to a tertiary treatment mechanism with UV lamps to remove pathogens before being reused for lawn irrigation and toilet flushing. Samples from influent and effluent were analyzed for various physical, chemical, and microbiological characteristics. The degreaser significantly reduced turbidity, TSS, total BOD5, and total COD levels in greywater. When combined with the green wall façades, the system demonstrated high removal efficiencies, particularly for turbidity, TSS, total COD, and total BOD5. The treated effluent met irrigation reuse standards for all the parameters, including total coliform and E. coli concentrations. The UV disinfection unit proved to be an effective post-treatment step, ensuring that water quality standards for reuse were met. The system’s overall performance highlights its ability to manage low- to medium-strength greywater. Results suggest the applied green wall system has significant potential for wider adoption in urban settings. Full article
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39 pages, 386 KB  
Article
Higher-Order Expansions for Estimators in the Presence of Nuisance Parameters
by Paul Rilstone
Mathematics 2025, 13(2), 179; https://doi.org/10.3390/math13020179 - 7 Jan 2025
Viewed by 1038
Abstract
Higher-order asymptotic methods for nonlinear models with nuisance parameters are developed. We allow for both one-step estimators, in which the nuisance and parameters of interest are jointly estimated; and also two-step (or iterated) estimators, in which the nuisance parameters are first estimated. The [...] Read more.
Higher-order asymptotic methods for nonlinear models with nuisance parameters are developed. We allow for both one-step estimators, in which the nuisance and parameters of interest are jointly estimated; and also two-step (or iterated) estimators, in which the nuisance parameters are first estimated. The properties of the former, although in principle simpler to conceptualize, are more difficult to establish explicitly. The iterated estimators allow for a variety of scenarios. The results indicate when second-order considerations should be taken into account when conducting inferences with two-step estimators. The results in the paper accomplish three objectives: (i) provide simpler methods for deriving higher-order moments when nuisance parameters are present; (ii) indicate more explicitly the sources of deviations of estimators’ sampling distributions from that given by standard first-order asymptotic theory; and, in turn, (iii) indicate in which situations the corrections (either analytically or by a resampling method such as bootstrap or jackknife) should be made when making inferences. We illustrate using several popular examples in econometrics. We also provide a numerical example which highlights how a simple analytical bias correction can improve inferences. Full article
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