Residual-based efficient and powerful independence testing in multivariate isotonic semiparametric nonlinear regression

Abstract

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.