Estimating Optimal Transformations for Multiple Regression and Correlation

Abstract

In regression analysis the response variable Y and the predictor variables X ₁ …, X_p are often replaced by functions θ(Y) and Ø₁(X ₁), …, Ø _p (X_p ). We discuss a procedure for estimating those functions θ and Ø₁, …, Ø _p that minimize e ² = E{[θ(Y) — Σ Ø _j (X_j )]²}/var[θ(Y)], given only a sample {(y_k , x_k1 , …, x_kp ), 1 ⩽ k ⩽ N} and making minimal assumptions concerning the data distribution or the form of the solution functions. For the bivariate case, p = 1, θ and Ø satisfy ρ = p(θ, Ø) = max_θ,Øρ[θ(Y), Ø(X)], where ρ is the product moment correlation coefficient and ρ is the maximal correlation between X and Y. Our procedure thus also provides a method for estimating the maximal correlation between two variables.

Key Words:

• Smoothing
• ACE