UNIFORM CONVERGENCE OF SERIES ESTIMATORS OVER FUNCTION SPACES

This paper considers a series estimator of <bold>E</bold>[α(<italic>Y</italic>)|λ(<italic>X</italic>) = λ̄], (α,λ) null 𝛢 × Λ, indexed by function spaces, and establishes the estimator's uniform convergence rate over λ̄ null <bold>R</bold>, α null 𝛢, and λ null Λ, when 𝛢 and Λ have a finite integral bracketing entropy. The rate of convergence depends on the bracketing entropies of 𝛢 and Λ in general. In particular, we demonstrate that when each λ null Λ is locally uniformly null₂-continuous in a parameter from a space of polynomial discrimination and the basis function vector <italic>p</italic>^null in the series estimator keeps the smallest eigenvalue of <bold>E</bold>[<italic>p</italic>^null(λ(<italic>X</italic>))<italic>p</italic>^null(λ(<italic>X</italic>))‼] above zero uniformly over λ null Λ, we can obtain the same convergence rate as that established by de Jong (2002, <italic>Journal of Econometrics</italic> 111, 1–9).