Aug 2026· Mathematics· Vol 14, pp. 2935· 0 citations· 10 references
Abstract
We introduce a robust nonparametric regression framework for functional covariates that combines functional principal component analysis (FPCA), marginal copula-scale normalization, bounded-score M-estimation, and multivariate Bernstein smoothing. The proposed procedure reduces the infinite-dimensional functional predictor to a low-dimensional score representation, transforms the retained scores onto the compact unit cube, and estimates a conditional M-functional through a smoothly aggregated system of local estimating equations. This construction is designed to accommodate nonlinear regression structure, heavy-tailed score distributions, and response contamination while limiting the influence of extreme observations. Under suitable regularity and undersmoothing conditions, we establish pointwise and uniform consistency, derive explicit convergence rates, and prove asymptotic normality. The limiting variance contains an explicit Bernstein concentration factor that plays a role analogous to the integrated squared kernel in classical nonparametric regression. The analysis also clarifies the interaction among the projection dimension, the Bernstein resolution, the empirical copula transformation, and the effective local sample size. The finite-sample performance of the method is examined through simulations involving heavy-tailed functional scores, Student-t errors, nonlinear regression effects, and increasing response contamination. The proposed estimator exhibits strong overall predictive performance and good robustness, with particularly favorable behavior under absolute-error criteria.
We study estimation of the p*p residual scatter (shape) matrix in a high-dimensional multivariate linear regression, where p and n grow proportionally. When the coefficient matrix obeys a known linear restriction of rank q<d, as in multivariate analysis of variance, growth-curve models, and reduced-rank regression, the restricted fit leaves additional residual degrees of freedom that sharpen estimation of the shape matrix. To accommodate heavy-tailed errors, we work with independent elliptically distributed rows under a mild scale condition, a finite second moment on the radii, which is far weaker than the usual sub-Gaussian assumptions and covers every multivariate-t law with more than two degrees of freedom. Shrinking the restricted residual sample covariance directly is unsound here, since its limiting spectrum depends on the radial distribution. We instead shrink a scale-invariant scatter of the restricted residuals, whose spectrum is distribution-free over the elliptical family and obeys the same limiting law as under Gaussian errors, at a smaller effective aspect ratio. The resulting estimator attains the rotation-equivariant oracle and is asymptotically optimal within that class, and a Stein-type combination with the unrestricted estimator dominates it while remaining safe under misspecification. We further correct for the case in which the restriction is itself selected from the data. Simulations, a growth-curve experiment, and two real-data analyses illustrate the results.
H. Karamikabir, Mohammad Arashi Department of Statistics, Faculty of Intelligent Systems Engineering et al.· 0 citations
We study residual-based independence testing in multivariate isotonic semiparametric nonlinear regression models, where the regression function combines a finite-dimensional nonlinear parametric component with an infinite-dimensional shape-constrained isotonic component. A key assumption is the independence between regression errors and joint covariates, whose violation may indicate model misspecification or hidden dependence. To assess this assumption, we propose two residual-based nonparametric diagnostic procedures based on distance covariance and the Bergsma–Dassios τ∗statistic. The former detects general nonlinear dependence, while the latter provides a rank-based robust alternative. Estimation is performed using an alternating least squares algorithm that combines nonlinear least squares and multivariate isotonic regression. We establish convergence of the algorithm, derive joint asymptotic representations for the estimators, and show that the plug-in effect of estimated residuals is asymptotically negligible. The asymptotic behavior of the tests is investigated under the null hypothesis and contiguous local alternatives, and large-sample power functions, along with Pitman and Bahadur efficiencies, are obtained. Simulation studies demonstrate accurate size control and strong power under a variety of dependence structures, including heavy-tailed, heteroscedastic, and contaminated settings. Distance covariance generally exhibits higher sensitivity under smooth nonlinear alternatives, whereas τ∗shows greater robustness to outliers and heavy-tailed errors. A real-data analysis using the Boston Housing dataset illustrates the practical applicability of the proposed methodology. The resulting framework provides a theoretically grounded and computationally efficient approach for model adequacy assessment in shape-constrained semiparametric nonlinear regression.
Sthitadhi Das· Hacettepe Journal of Mathema...· 0 citations
Laplace factor models (LFMs) provide a heavy-tailed alternative to Gaussian factor models by representing high-dimensional observations through a low-rank common component and Laplace-distributed idiosyncratic errors. This paper develops an assumption-consistent finite-sample analysis of matrix concentration, covariance estimation, and Monte Carlo integration under this model. We first formulate the model with explicit dimensional, independence, covariance, and identifiability conditions. Standard matrix Laplace-transform and matrix Bernstein inequalities are then recalled with their precise applicability conditions. Because untruncated Laplace variables are neither almost surely bounded nor strongly log-concave, these standard results cannot be applied directly in the forms commonly used for bounded or Gaussian-like observations. To address this issue, we analyze a coordinatewise truncated covariance estimator and derive an operator-norm bound that separates the stochastic estimation error from the truncation bias. The resulting rate depends on the effective rank and the logarithm of the ambient dimension and is therefore not dimension-free. For Monte Carlo integration, we replace strong-log-concavity arguments by a sub-exponential concentration analysis that is compatible with independent Laplace errors and yields non-asymptotic absolute- and relative-error bounds. Simulation studies compare empirical tails with the classical matrix Bernstein bound, evaluate ordinary, truncated, winsorized, PCA, POET-type, and Huberized covariance estimators, and we compare Laplace-based and Studentized confidence intervals. The results show that the classical Bernstein bound can be conservative, and truncation involves a substantial bias–variance trade-off. In a Wine chemical-analysis application, three factors explain 66.53% of the standardized variance, and POET-type covariance estimation attains a cross-validated balanced accuracy of 0.9901. These findings clarify both the scope and the limitations of finite-sample analysis for LFMs.
Siqi Liu, X. Wen, A. Adekpedjou et al.· Mathematics· 0 citations
Classical canonical correlation analysis becomes numerically unstable when the number of variables is large relative to the sample size and is sensitive to contamination in observations or individual cells. This study develops an integrated robust and regularized procedure that combines bounded cellwise wrapping, shrinkage estimation of the joint correlation matrix, and robust reweighting in a low-dimensional canonical score space. The resulting observation weights enter a second regularized canonical correlation fit, so the final estimator remains well defined when the combined number of variables exceeds the sample size. The simulation study shows that relative estimation accuracy depends on the signal strength, contamination mechanism, and dimensional configuration. The proposed estimator is competitive in several moderate-signal settings and has a clear computational advantage, whereas the minimum regularized covariance determinant plug-in estimator provides lower estimation error in many high-signal configurations. An additional ultra-high-dimensional experiment demonstrates numerical feasibility with modest memory use but also reveals substantial attenuation, identifying a limitation of the present dense estimator. The results therefore support a regime-dependent interpretation rather than a claim of uniform superiority. The complete reproducible simulation workflow is provided.
Hasan Bulut, Müjgan Zobu, V. Saglam· Mathematics· 0 citations
A robust tensor quantile regression method, in which CANDECOMP/PARAFAC (CP) decomposition is employed for dimension reduction, and an exponential‐type penalty (ETP) is imposed at the element‐wise level to achieve sparse variable selection.
Tan Meng, Shuo Liu, Mao-Zai Tian· Statistical analysis and dat...· 0 citations
We develop a theory of nonlinear shrinkage covariance estimation for nonparanormal (Gaussian-copula) models, in which each observed coordinate is an unknown strictly increasing transformation of a latent Gaussian vector. This model accommodates arbitrary marginal skewness and heavy marginal tails while retaining a Gaussian dependence structure, and it is the natural semiparametric setting for heavy-tailed, asymmetric financial returns. Our estimator, marginal-free nonlinear shrinkage (MENS), applies an oracle nonlinear shrinkage function to the eigenvalues of the normal-scores rank-covariance matrix. We give the almost-sure convergence of the empirical spectral distribution of the normal-scores covariance to the generalized Marchenko-Pastur law of Sigma, and asymptotic optimality of MENS among rotation-equivariant estimators under Frobenius loss. We establish a Baik-Ben Arous-Peche phase transition for spiked latent correlations. The MENS attains the robustness of rank-based estimation and the efficiency of nonlinear shrinkage at once within this class. We corroborate the theory with a simulation study that isolates the marginal-invariance property and the spiked transition. In an out-of-sample minimum-variance backtest on S&P 500 stocks, MENS delivers a better-conditioned covariance estimate, lower realized portfolio volatility, and lower turnover than linear shrinkage, illustrating its practical value for high-dimensional allocation and decision-making.
H. Karamikabir, Mohammad Arashi Department of Statistics, Faculty of Intelligent Systems Engineering et al.· 0 citations
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