The analysis of high-dimensional count data presents a unique set of challenges, including overdispersion, zero-inflation, and complex nonlinear relationships that traditional generalized linear models and standard machine learning approaches often fail to adequately address. This study introduces and validates a novel Random Forest framework specifically developed for high-dimensional Poisson and Negative Binomial regression, designed to overcome the limitations of existing methods. Through comprehensive simulations and a real-world genomic application to the Norwegian Mother and Child Cohort Study, we demonstrate that the proposed methods achieve superior predictive accuracy, quantified by lower root mean squared error and deviance, and critically produced exceptionally stable and interpretable feature selections. Our theoretical and empirical results show that these distribution-optimized ensembles significantly outperform both penalized-likelihood techniques and naive-transformation-based ensembles in balancing statistical robustness with biological interpretability. The study concludes that the proposed frameworks provide a crucial methodological advancement, offering a powerful and reliable tool for extracting meaningful insights from complex count data in fields ranging from genomics to public health.

It has long been understood that there exists a strong association between a student’s belief in the future utility of mathematics and their self-efficacy in mathematics. This study re-examines this association by studying these variables based on data collected from a sample of 21,444 ninth-grade students who participated in the 2009 High School Longitudinal Study (HSLS09). The nature of the association between future utility beliefs of students in mathematics and self-efficacy of students in mathematics is explored visually using the simple correspondence analysis technique. The main feature that will be utilised from this technique is a two-dimensional graphical display, referred to as a correspondence plot. By studying the HSLS09 data, the first two axes of such a plot summarised nearly 99% of the statistically significant association that exists between a student’s beliefs in the future utility of mathematics and their mathematics self-efficacy. It is shown visually that students who strongly believe in the future importance of studying mathematics also perform strongly in the subject, while those who do not believe that there is any future utility from studying mathematics do not perform well at it. This study, therefore, suggests that mathematics ability is associated with a student’s perception of its future importance.

Ensemble methods have proven highly effective in enhancing predictive performance by combining multiple models. We introduce a novel ensemble approach, the Random Generalized Additive Logistic Forest (RGALF), which integrates generalized additive models (GAMs) within a random forest framework to improve binary classification tasks. Unlike traditional random forests, which rely on piecewise constant predictions in terminal nodes, RGALF fits GAM logistic regression (LR) models to the data in each terminal node, enabling it to capture complex nonlinear relationships and interactions among predictors. By aggregating these node-specific GAMs, RGALF addresses multicollinearity, enhances interpretability, and achieves superior bias–variance tradeoffs, particularly in nonlinear settings. Theoretical analysis confirms that RGALF achieves Stone’s optimal rates for additive models ($\mathcal{O}(n^{-2k/(2k+d)}$) under appropriate conditions, outperforming the slower convergence of traditional random forests ($\mathcal{O}\left(n^{-2/3}\right)$). Furthermore, empirical results demonstrate RGALF’s effectiveness across both simulated and real-world datasets. In simulations, RGALF demonstrates superior performance over random forests (RFs), reducing variance by up to 69% and bias by 19% in nonlinear settings, with significant MSE improvements (0.032 vs. RF’s 0.054 at $n=1000$), while achieving optimal convergence rates ($\mathcal{O}\left(n^{-0.48}\right)$ vs. RF’s $\mathcal{O}\left(n^{-0.29}\right)$). On real-world medical datasets, RGALF attains near-perfect accuracy and AUC: 100% accuracy/AUC for Heart Failure and Hepatitis C (HCV) prediction, 99% accuracy/100% AUC for Pima Diabetes, and 98.8% accuracy/100% AUC for Indian Liver Patient (ILPD), outperforming state-of-the-art methods. Notably, RGALF captures complex biomarker interactions (BMI–insulin in diabetes) missed by traditional models.

Multicollinearity poses significant challenges to parameter estimation in regression models, often undermining the reliability of traditional methods like maximum likelihood estimation (MLE). This study addresses the issue by evaluating and enhancing regularization techniques, specifically the ridge-based regression (RBR), the Liu regression estimator (LRE), and a modified two-parameter ridge estimator (MTPRE) within the context of the Beta regression model (BRM). However, the selection of appropriate shrinkage parameters remains a persistent challenge. To address this limitation, we propose an MTPRE that eliminates the need for shrinkage parameter tuning, thereby improving estimation stability and accuracy. Through extensive simulation studies, the MTPRE consistently outperformed the MLE, RBR, and LRE under severe multicollinearity based on the mean squared error (MSE). The effectiveness of proposed estimators was further validated using a real-world gasoline yield dataset having multicollinearity issues, where the MTPRE demonstrated superior predictive accuracy and estimation precision. These results highlight the potential of the MTPRE as a practical and efficient method for handling multicollinearity in regression analysis.

Fractional cointegration has been extensively examined in time series analysis, but its extension to heterogeneous panel data with unobserved heterogeneity and cross-sectional dependence remains underdeveloped. This paper develops a robust framework for testing fractional cointegration in heterogeneous panel data, where unobserved heterogeneity, cross-sectional dependence, and persistent shocks complicate traditional approaches. We propose the Bayesian Tapered Narrowband Least Squares (BTNBLS) estimator, which addresses three critical challenges: (1) spectral leakage in long-memory processes, mitigated via tapered periodograms; (2) precision loss in fractional parameter estimation, resolved through narrowband least squares; and (3) unobserved heterogeneity in cointegrating vectors (${\theta }_i$) and memory parameters ($\nu ,\delta$), modeled via hierarchical Bayesian priors. Monte Carlo simulations demonstrate that BTNBLS outperforms conventional estimators (OLS, NBLS, TNBLS), achieving minimal bias (0.041–0.256), near-nominal coverage probabilities (0.87–0.94), and robust control of Type 1 errors (0.01–0.07) under high cross-sectional dependence ($\rho =0.8$), while the Bayesian Chen–Hurvich test attains near-perfect power (up to 1.00) in finite samples. Applied to Purchasing Power Parity (PPP) in 18 fragile Sub-Saharan African economies, BTNBLS reveals statistically significant fractional cointegration between exchange rates and food price ratios in 15 countries ($p<0.05$), with a pooled estimate ($\widehat{\theta }=0.33$, $p<0.001$) indicating moderate but resilient long-run equilibrium adjustment. These results underscore the importance of Bayesian shrinkage and spectral tapering in panel cointegration analysis, offering policymakers a reliable tool to assess persistence of shocks in institutionally fragmented markets.

Evaluating whether clinical classifiers perform equitably across patient subgroups is a central requirement for the responsible deployment of machine learning in medicine. Conventional approaches test one fairness metric at a time, such as sensitivity, positive predictive value, or area under the receiver operating characteristic curve, and therefore cannot detect disparities that manifest only in the joint structure of a group-specific confusion matrix. We develop a unified hypothesis-testing framework rooted in random matrix theory that compares demographic groups through the $L^2$ distance between their joint eigenvalue densities, yielding a scalar spectral divergence that is sensitive to every cell of the $2\times 2$ confusion matrix simultaneously. We derive the closed-form spectral divergence for Gaussian-approximated eigenvalue densities, prove almost-sure consistency of the empirical estimator via the delta method, and construct an extreme-value (Gumbel) test statistic with family-wise error rate control. Monte Carlo experiments comprising $\mathrm{10,000}$ replications across balanced, moderately imbalanced, and severely imbalanced group-size regimes show that the spectral test keeps Type I errors close to its nominal level while achieving power exceeding $90\%$ in complex and multi-dimensional violation scenarios, where the best single-metric competitor reaches at most $63\%$. Three clinical benchmark datasets from the UCI Machine Learning Repository utilised include Pima Indians Diabetes ($n=768$), Cleveland Heart Disease ($n=303$), and Heart Failure Clinical Records ($n=299$). Results confirm that the spectral method detects statistically significant ($p<0.001$) performance disparities missed by all three conventional tests. These results support eigenvalue-based divergence as a practical, model-agnostic diagnostic equity tool for clinical machine learning audits.