Peter Hall constructs a tractable mathematical model for kernel-based projection pursuit regression approximation, providing explicit formulae for the bias and variance of estimators. He demonstrates that the bias of an orientation estimate dominates the error about the mean, which is asymptotically negligible compared to the bias. However, both bias and error about the mean are of the same order for projection pursuit curve estimates. The article discusses the implications of these formulae for bias and variance. Hall also proves that the common form of kernel-based projection pursuit regression estimates projections with convergence rates similar to those in one-dimensional problems, but requires an extra derivative to achieve this. He suggests a two-stage procedure to improve the convergence rate of orientation estimators without altering the convergence rate of curve estimates, though it exacerbates the numerical problem of multiple minima in the orientation function. The article is structured to describe the model, develop essential calculus, state main results, and provide proofs.Peter Hall constructs a tractable mathematical model for kernel-based projection pursuit regression approximation, providing explicit formulae for the bias and variance of estimators. He demonstrates that the bias of an orientation estimate dominates the error about the mean, which is asymptotically negligible compared to the bias. However, both bias and error about the mean are of the same order for projection pursuit curve estimates. The article discusses the implications of these formulae for bias and variance. Hall also proves that the common form of kernel-based projection pursuit regression estimates projections with convergence rates similar to those in one-dimensional problems, but requires an extra derivative to achieve this. He suggests a two-stage procedure to improve the convergence rate of orientation estimators without altering the convergence rate of curve estimates, though it exacerbates the numerical problem of multiple minima in the orientation function. The article is structured to describe the model, develop essential calculus, state main results, and provide proofs.