Addressing incomplete and non-normally distributed multivariate data poses significant challenges in medical research, particularly when the interest is in discovering underlying data structures. This article introduces a robust factor analysis framework for handling missing data by employing the canonical fundamental skew- factor analysis (CFUSTFA) model, which incorporates the canonical fundamental skew- distribution into the latent factors and error terms. This versatile framework accounts for skewness, heavy tails, and missing data, thereby enhancing the model's ability to capture complex structures commonly observed in biomedical datasets. For parameter estimation under the missing at random mechanism, we develop a computationally efficient alternating expectation-conditional maximization algorithm within the maximum likelihood framework. This approach facilitates the simultaneous imputation of missing values and the extraction of low-dimensional factor representations. Standard errors for parameter estimates are also derived using a general information matrix-based approach. The proposed methodology is validated through simulations and applied to a hepatitis C virus laboratory dataset exhibiting skewness, excess kurtosis, and missingness. Our findings highlight the capability of the CFUSTFA model to robustly capture complex, incomplete, and asymmetric biomedical data, offering enhanced inference and interpretability compared with existing factor analysis approaches.

During spacecraft launch, flight, and docking operations, monitored images often suffer from spatially variant blur caused by atmospheric turbulence, defocusing, and relative motion. To address this challenge, we propose a novel unsupervised deblurring framework tailored specifically for spacecraft imagery. Our approach incorporates three key innovations: First, we design a detail-preserving local region selection strategy based on multi-scale morphological gradients with adaptive thresholding, which optimizes regions for blur kernel estimation. Second, we define a blur kernel error term and integrate it into the degradation model, introducing explicit error correction constraints into the alternating iterative minimization process. Third, we incorporate Shearlet transform regularization to enhance the recovery of fine local details. Experimental results demonstrate that our method significantly outperforms state-of-the-art unsupervised techniques and even surpasses several advanced deep learning approaches in preserving complex structural details under spatially variant degradation. Our code and data are available at https://github.com/bsfsf/Image_deblur.

Big data fundamentally differs from traditional data, characterized by large volumes, high rates of missing values, and a lack of conformity to the normal distribution assumption for independent variables in traditional regression models. Consequently, there is a pressing need for new methodologies to enhance estimation accuracy and computational efficiency. Considering the hierarchical characteristics of data structure, we extend composite quantile regression (CQR) to accommodate hierarchical data assumptions and propose a hierarchical composite quantile regression (HCQR) model. In case of missing response variables, the regression coefficients are decomposed into individual and common components, where the inverse probability weighting is utilized for imputation. To enhance computational efficiency and achieve more accurate parameter estimates, we optimize the constructed function using the majorization-minimization algorithm. Numerical simulations of the proposed model under various missing rates reveal that our method not only improves parameter estimation accuracy but also effectively addresses non-normally distributed error terms. Finally, we apply the model to predict pancreatic cancer incidence, which provides valuable reference for the prediction and prevention of pancreatic cancer in clinical practice.

Stochastic frontier models have attracted considerable attention due to the incorporation of an inefficiency term in addition to the conventional error term. In this paper, we propose a general estimation framework for panel stochastic frontier models that accommodates potential heterogeneity through latent group structures. The framework is tailored to the distinctive features of stochastic frontier models and is paired with a practical hybrid estimation procedure that combines individual-level and joint panel estimation. We illustrate the estimation framework using a panel stochastic frontier model that treats the inefficiency term as a random effect, and show that it can be readily extended to a range of fixed effects specifications common in the literature. Simulation studies indicate strong finite-sample performance, and we further demonstrate the practicality of the approach in an empirical application to the cost efficiency of the U.S. commercial banking sector.

A low-pass filter is essential for stabilizing strictly proper repetitive-control systems, but it inevitably degrades steady-state tracking accuracy due to gain attenuation and phase lag. This article presents a new filter design and optimization method that improves both transient response and steady-state accuracy in continuous-time repetitive-control systems. First, the gain and phase characteristics of conventional filter-based repetitive controllers are rigorously analyzed to reveal the relationship between filter parameters and tracking performance. Based on this analysis, a new filter structure is designed to precisely compensate for gain attenuation and phase delay, specifically at the fundamental frequency, by minimizing the error term without increasing the filter bandwidth. A guideline for selecting the filter parameters for varying periodic trajectories is also provided. In addition, according to the one-to-one mapping between control and learning behaviors and their respective gains, dual performance indices are constructed to account for tracking error and control effort across multiple learning cycles. A multiobjective optimization framework is then developed to directly tune these gains subject to stability constraints, achieving an optimal balance between rapid transient convergence and control energy efficiency. Experimental results validate the effectiveness and superiority of the design.

A fundamental requirement for full autonomy for mobile robots is accurate navigation even in situations where satellite navigation or cameras are unavailable. In such practical scenarios, relying only on inertial sensors will result in navigation solution drift due to inherent noise and error terms. One of the emerging solutions to mitigate drift is to maneuver the robot in a snake-like slithering motion to increase the inertial signal-to-noise ratio, allowing the regression of the mobile robot position. In this work, MoRPI-PINN, a physics-informed neural network framework, has been proposed for inertial-based mobile robot navigation. These investigations are crucial for gaining deeper insights into the mobile robot's pure inertial navigation solution. By embedding physical laws and constraints into the training process, MoRPI-PINN provides an accurate and improved navigation solution. Using real-world experiments, we show accuracy improvements of over 80% compared to other baseline approaches for unseen trajectories. Moreover, MoRPI-PINN is a lightweight approach that can be implemented even on edge devices and used in any typical mobile robot application.

Using a recent method developed by Mahatab, we obtain an improved $\Omega$-bound for the error term arising in lattice counting problem of bodies of revolution in $\mathbb R^3$ around a coordinate axis and having smooth boundary with bounded nonzero curvature. This strengthens an earlier result by K\"uhleitner and Nowak.

Functional connectivity varies across individuals due to genetic and environmental factors, yet classical twin models typically confound non-shared environment with measurement error and are largely limited to resting-state analyses. We hypothesized that: i) explicitly modeling measurement error from repeated fMRI sessions enables more accurate application of classical twin models (ACE/ADE) to functional connectivity; ii) model applicability depends on scan-length and parcellation granularity; iii) genetic and environmental effects on functional connectomes show differentiated functional modules across conditions. We extended ACE/ADE models to include a repeated-scan derived error term by analyzing monozygotic and dizygotic twins from the Young-Adult Human Connectome Project dataset. Genetic and environment variance components were estimated for all functional couplings across resting-state and task conditions, integrated across conditions using a minimum-error criterion, and analyzed using multilayer community detection across resolution scales. Functional couplings segregated into distinct categories characterized by shared environmental, additive, dominant, or epistatic influences, with a substantial fraction not meeting twin-model assumptions. Integrating across conditions revealed hierarchical community structure in genetic and environmental components observed across community resolution scales. Incorporating measurement error into twin models improves interpretability and applicability at the functional connectome level, revealing that genetic and environmental influences are structured into coherent, multiscale brain networks.

Unemployment and energy poverty are significant social challenges faced by many countries, including OPEC members. The ongoing global transition to renewable energy has exacerbated unemployment in OPEC countries due to reduced reliance on traditional energy sources, highlighting the need for a comprehensive study of unemployment and energy poverty in these regions. This study will cover the period from 2015 to 2023, utilizing secondary data from all 12 OPEC member countries. Both trend and econometric tests are employed. The study employs the System Generalized Method of Moments (GMM) for panel data estimation, recognized for its efficiency in addressing endogeneity, unobserved heterogeneity, and dynamic relationships. The study validates the reliability of the model with the Arellano-Bond test for autocorrelation to assess the independence of error terms. The findings reveal that access to electricity significantly increases unemployment, likely due to automation and limited job absorption. Conversely, access to cooking gas shows a potential to reduce unemployment, though the effect is not statistically strong. The study recommends aligning energy strategies with employment goals, investing in vocational training, empowering women through clean cooking initiatives, and adopting inclusive, gender-responsive policies to ensure equitable benefits from energy transitions.

Simultaneous Equation Models (SEMs) represent an essential category of statistical models where dependent variables are determined not only by independent variables but also by other dependent variables within the system. This characteristic implies that the explanatory variables may be interrelated with the dependent variables, reflecting equilibrium mechanisms commonly found in economic models. For example, in a standard supply and demand model, both the quantity supplied and the quantity demanded are influenced by the market price. However, it is also possible for producers to adjust their prices based on observed consumer demand, illustrating the bidirectional relationships in SEMs. The presence of Multicollinearity and Heteroscedasticity in a model is often problematic during estimation, especially when the system of equations is complex, as is the case of SEM. This paper provides a detailed literature review under the linear regression model on the two assumption violators (Multicollinearity and Heteroscedasticity). Multicollinearity, a situation where explanatory variables in a regression model are highly correlated, remains a fundamental challenge in statistical analysis, particularly when dealing with complex datasets. When multicollinearity occurs, it becomes difficult to isolate the individual effects of predictor variables, often leading to inflated standard errors, unstable parameter estimates, and reduced statistical power. Another significant challenge in estimating linear regression model is the presence of heteroscedasticity, where the variance of the error terms is not constant across observations. Heteroscedasticity violates one of the key assumptions of the OLS method, leading to inefficient and biased estimates of the regression parameters. To address this issue, heteroscedasticity-consistent estimators, such as White's robust standard errors and the Generalized Method of Moments (GMM), have been developed (Pérez-Sánchez et al., 2021). These methods provide more reliable estimates in the presence of heteroscedasticity by accounting for the varying error variances. However, when Multicollinearity and Heteroscedasticity co-exist in the model (whether a linear or simultaneous equation model), estimation is usually very cumbersome.

This study proposes a semiparametric approach to Bayesian linear regression using the Bayesian generalized method of moments. Unlike conventional methods, the proposed approach does not require specifying a probability distribution for the error term and avoids relying on the assumptions of homoskedasticity and normality. The primary advantage of this method is its ability to provide valid inference, particularly credible intervals with correct coverage, even in the presence of heteroskedasticity. Simulation studies show that both frequentist and Bayesian methods assuming homoskedasticity yield confidence or credible intervals with poor coverage under heteroskedasticity, whereas the proposed method consistently achieves accurate and reliable uncertainty quantification. A case study further demonstrates that failing to account for heteroskedasticity can lead to misleading conclusions. Overall, the proposed method offers a robust and practical alternative to conventional likelihood-based Bayesian linear regression, with potential extensions to more complex models involving linear components. (PsycInfo Database Record (c) 2026 APA, all rights reserved).

Abstract We establish an asymptotic formula for counting integer solutions with smooth weights to an equation of the form , where is a non‐zero integer, with an explicit main term and a strong bound on the error term in terms of the size of the variables as well as of . We also establish an asymptotic formula for counting integer solutions with smooth weights to the congruence , where is a large prime, with a strong bound on the error term.

Asymptotic of the plane overpartition with explicit error terms

Abstract This article deals with the time trend evolution of US natural catastrophes over the last forty years. However, instead of using classical methods that impose an I(0) structure for the error term, we allow for potential long memory features by using fractional integration. Count and costs series are investigated for the US as a whole but also using disaggregated data by climate regions. The results indicate the presence of time trends in the majority of the cases examined and evidence of long memory is also found in a number of climate regions.

Abstract Recent work by M. Afifurrahman established the first asymptotic estimates with error terms for the number of $2\times 2$ matrices with fixed non-zero determinant $n\in\mathbb{N}$ , and with coefficients bounded in absolute value by X . In this paper we present a new proof of this result, which also gives an improved error term as $X\rightarrow\infty$ . Similar to Afifurrahman’s result, our error term is uniform in both n and X , and our estimates are significant for X as small as $n^{1/2+\delta}$ . To complement this, we also demonstrate that the exponent $1/2+\delta$ in this statement cannot be reduced, by establishing a result which gives a different asymptotic main term when n is either a prime or the square of a prime, and when $X=n^{1/2}$ .

The Grassmann neural network (GrNet) has shown strong capability in modeling nonlinear visual data. However, its performance is often limited by information degradation caused by multi-level data compression and insufficient discriminative learning. To address these issues, we propose a U-shaped Grassmann manifold network, which extends GrNet with an encoder–decoder architecture supervised by a reconstruction error term to preserve semantic transformations. Skip connections based on Grassmannian geometries are further introduced to enhance information flow between the corresponding encoding and decoding layers. In addition, a metric learning regularization term based on Grassmannian distance is incorporated into the loss function to enhance the discriminative capability. Extensive experimental results on multiple visual classification tasks demonstrate the effectiveness of the proposed method.

On the error term of the sum involving floor function

Somatic items used in depression assessments can potentially overlap with symptoms related to physical illness, including systemic sclerosis (SSc). No studies have looked at whether somatic depression items may be influenced by diffuse versus limited SSc disease subtypes, which are associated with varying degrees of symptom presentation. The objective of this study was to evaluate differential item functioning (DIF) in items of the 8-item Patient Health Questionnaire (PHQ-8) across SSc subtypes. We also assessed the PHQ-8 for DIF across language (English and French), sex, and age. Participants enrolled in the Scleroderma Patient-Centered Intervention Network Cohort who completed the PHQ-8 at enrollment between April 2014 and October 2020 were included. Confirmatory factor analysis (CFA) was used to evaluate the unidimensional structure of the PHQ-8, and DIF analyses based on SSc subtype, language, sex, and age were conducted using Multiple Indicators Multiple Causes models. In total, 2,191 participants were included. CFA with several covarying error terms supported a one-factor structure for the PHQ-8 (Tucker-Lewis Index=0.99, Comparative Fit Index=0.98, Root Mean Square Error of Approximation=0.08). We did not identify statistically significant DIF based on SSc subtype. Statistically significant DIF was found in one item for language, one item for sex, and two items for age. However, the effect of DIF on overall PHQ-8 scores was negligeable in all cases. We did not find evidence that the PHQ-8 performs differently across SSc subtypes, language of administration, sex, and age groups.

ttScreening (TT) is an effective high-dimensional screening algorithm to identify important cytosine-phosphate-guanine dinucleotide (CpG) sites associated with DNA methylation. Via simulations, we aimed to examine the impact of influential outliers on TT’s performance. We simulated K = 2,000 and 10,000 CpG sites across n = 100 and 200 subjects, linearly associated with a continuous outcome, $$x_1$$, and other latent variables with the error term following a normal or Cauchy distribution. Among the K CpGs, 10 were associated with $$x_1$$ (informative CpGs) while the remaining sites were not associated with $$x_1$$ (non-informative CpGs). We artificially created 1 to 5 influential points in one informative and one non-informative CpG site and compared TT’s accuracy to Bonferroni and false discovery rate (FDR)-based approaches. TT performed as well as or better than the FDR and Bonferroni-based approaches, across all degrees of influentiality. When focusing on non-informative CpG detection, regardless of sample size, all approaches had high accuracy (above 85%, overall) at their optimal cutoff for a single influential point. Among the CpG sites with a higher number of influential points (five points) and a normal error term, TT required a minimum cutoff of 70% for accuracy $$>0\%$$ compared to FDR and Bonferroni, both of which had an accuracy of 0% for n = 100 and 200. However, increasing the TT cutoff to 80% increased accuracy to 20% and 24%, respectively, and further increased to 97% and 99% for a 90% cutoff, respectively (among K = 2000). We observed the same patterns for 10,000 CpGs and informative CpG detection. When Cauchy error terms were applied, the same patterns held, but with a higher magnitude of accuracy for all approaches, and thus TT required lower cutoffs to achieve 100% accuracy. In summary, in the presence of influential data, we recommend a more conservative cutoff of 70–90% compared to the default cutoff of 50–70% suggested by Ray et al. (in Biomed Res Int 2016(1):2615348, 2016). TT, Bonferroni, and FDR are capable approaches for type 1 protection when screening high-dimensional data. However, in the presence of influential data, TT is likely to be the most robust approach.

Differential light shift (DLS) is an important error term that limits the atom interferometer's measurement precision, especially for the case of the electro-optic modulator (EOM)-based scheme, where multiple laser sidebands exist, and their ratios are hard to control synchronously. This article carried out an experimental and theoretical study on this subject. By conducting long-term gravity measurement, we find that the gravity exhibits drifts of about 13.13 μGal, and is strongly correlated to the Raman laser's sidebands. A model of the DLS-induced gravity error is established and a DLS compensation method is proposed to suppress the gravity drift to 2.54 μGal. Besides the compensation method, we propose a Dual-Sideband Ratio Locking scheme to more robustly eliminate the gravity measurement drift. By feeding back to both the EOM microwave power and the tapered amplifier's temperature, this method locks both the ±1 order sideband to a stability level of 10-5, which corresponds to a gravity error of less than 0.1 μGal. Long-term gravity measurement is carried out after the locking method, showing a long-term stability of 1.6 μGal. The proposed methods will benefit the suppression of the DLS effect for high-precision atom interference measurement.