A revisit to Pearson correlation coefficient under multiplicative distortions

We consider the estimation of Pearson correlation coefficient when two continuous variables can not be directly observed but measured with multiplicative distortion measurement errors. Different from the identifiability conditions for the distortion functions in literature, we propose a new estimation method of Pearson correlation coefficient by transforming the linear model between two variables into varying coefficient models, a moment-based estimator of the Pearson correlation coefficient is proposed. This method can deal with the non-independence condition between the confounding variable and the unobserved variables. We study the asymptotic results of these proposed estimators, and we make some comparisons among the proposed estimators through the simulation. These methods are applied to analyze a real dataset for an illustration.