A consistent local linear estimator of the covariate adjusted correlation coefficient

Area of Science:

• Statistics
• Econometrics
• Biostatistics

Background:

• Estimating correlations between unobserved variables is crucial in various scientific fields.
• Observed data often undergo complex transformations, including multiplicative and additive effects from covariates.
• Existing methods may not adequately address dual transformations in correlation estimation.

Purpose of the Study:

• To develop a method for consistently estimating the correlation between two unobserved random variables (X, Y).
• To adjust for general dual additive and multiplicative effects of an observable covariate (U) on the unobserved variables.
• To provide a robust statistical framework applicable when direct observation of variables is not possible.

Main Methods:

• Utilizing observed data (X̃, Ỹ, U) where X̃ and Ỹ are transformed versions of X and Y.
• Employing non-parametric or semi-parametric estimation techniques to handle unknown smooth functions φ(l)(·) and ψ(l)(·).
• Developing a consistent estimator for the correlation coefficient under specified distributional assumptions.

Main Results:

• Demonstrated the consistency of the proposed estimation method.
• Showcased the ability to adjust for complex covariate effects (dual additive and multiplicative).
• Provided theoretical guarantees for the estimator's performance.

Conclusions:

• The developed method offers a reliable approach for correlation estimation with complex data transformations.
• This technique enhances the analysis of relationships between unobserved variables in the presence of covariate effects.
• The findings have implications for statistical modeling in fields relying on observational data.