UNIFORM CONVERGENCE OF SERIES ESTIMATORS OVER FUNCTION SPACES
研究了条件期望的系列估计量在函数空间上的一致收敛速度，该速度取决于函数空间的括号熵，并在特定条件下与已有结果一致。
This paper considers a series estimator of E [α( Y )|λ( X ) = λ̄], (α,λ) ∈ 𝛢 × Λ, indexed by function spaces, and establishes the estimator's uniform convergence rate over λ̄ ∈ R , α ∈ 𝛢, and λ ∈ 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 ∈ Λ is locally uniformly ℒ 2 -continuous in a parameter from a space of polynomial discrimination and the basis function vector p K in the series estimator keeps the smallest eigenvalue of E [ p K (λ( X )) p K (λ( X ))‼] above zero uniformly over λ ∈ Λ, we can obtain the same convergence rate as that established by de Jong (2002, Journal of Econometrics 111, 1–9).