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, 2

obtained as the root of the cubic score equation of a locally weighted Gaussian likelihood. 3

The estimator is nonparametric in the functional form of the correlation but assumes a 4

conditional bivariate Gaussian working model with standardised margins: pointwise con- 5

sistency requires only correctly specified conditional moments, whereas Gaussianity deter- 6

mines the asymptotic variance. Furthermore, we adopt a local linear smoothing technique 7

to correct the boundary effects, which are common in nonparametric estimation. We estab- 8

lish the consistency and asymptotic normality of the proposed estimator and demonstrate 9

its performance through simulation studies with non-Gaussian data. Additionally, we 10

apply our method to a single-participant resting-state MEG recording, estimating time- 11

varying power-envelope correlations in six frequency bands and comparing them with 12

static and sliding-window correlations.