Dependence and the dimensionality reduction principle

Abstract

Stone's dimensionality reduction principle has been confirmed on several occasions for independent observations. When dependence is expressed with φ-mixing, a minimum distance estimate θ̂n is proposed for a smooth projection pursuit regression-type function θ ∈, that is either additive or multiplicative, in the presence of or without interactions. Upper bounds on the L1-risk and the L 1-error of θ̂n are obtained, under restrictions on the order of decay of the mixing coefficient. The bounds show explicitly the additive effect of φ-mixing on the error, and confirm the dimensionality reduction principle.