Nonparametric least squares estimation of a regression function

Abstract

The problem 01 nonparametric function iittmg with me observation model: is considered, where are uncorrelated random variables irith zero mean and finite variance are fixed design variables diile / is an unknown function with only requirements imposed. AsymptoticN{n) ehaviour of the estimator is studied for obtained by the least' nuares methodbeing a complete set of orthonormal functions, Sufficient conditions for consistency in the sense of integrated and uniform mean square rror are derived. Bounds obtained for mean integrated square error (MISE) are used o indicate that for smooth functions the convergence rate of MISE is the optimal one.These results are obtained under minimal requirements imposed on measurement errors. Tnder additiona assumotion that these variables are mutually independent also strong niform converg once of the estimator is shown.