Nonparametric Correlation Estimator via Solving Cubic Equations and Its Application to Brain Functional Connectivity Analysis

In this paper, we propose a novel nonparametric estimator of the correlation coefficient, obtained as the root of the cubic score equation of a locally weighted Gaussian likelihood. The estimator is nonparametric in the functional form of the correlation but assumes a conditional bivariate Gaussian working model with standardised margins: pointwise consistency requires only correctly specified conditional moments, whereas Gaussianity determines the asymptotic variance. Furthermore, we adopt a local linear smoothing technique to correct the boundary effects, which are common in nonparametric estimation. We establish the consistency and asymptotic normality of the proposed estimator and demonstrate its performance through simulation studies with non-Gaussian data. Additionally, we apply our method to a single-participant resting-state MEG recording, estimating time-varying power-envelope correlations in six frequency bands and comparing them with static and sliding-window correlations.