On the Structural Distortion Induced by the Inverse Box–Cox Transformation

The Box–Cox transformation is widely used to induce approximate normality and linearity in statistical modelling. Within the Power Normal framework, it embeds non-Gaussian variables into a latent Gaussian structure where conditional relationships become linear. However, the inverse transformation does not generally preserve these functional relationships when returning to the original scale. In this paper, we formally analyze the discrepancy between the inverse image of the linear regression function in the transformed domain and the true conditional expectation in the original scale. We derive an explicit second-order decomposition showing that the conditional mean in the original scale consists of the inverse-transformed linear predictor plus a curvature-induced correction term proportional to the conditional variance. This distortion term depends explicitly on the transformation parameter and the local geometry of the inverse Box-Cox function. The analysis reveals that the loss of structural preservation under inversion is an intrinsic consequence of the nonlinear transformation and can be interpreted as a second-order Jensen-type correction. Numerical illustrations based on simulated bivariate Power Normal models confirm the theoretical findings. These results clarify a structural limitation of transformation-based Gaussian modelling and provide insight into its implications for statistical inference and applied modelling.