Joint Asymptotic Distribution of the Estimated Regression Function at a Finite Number of Distinct Points

As an approximation to the regression function m of Y on X based upon empirical data, E. A. Nadaraya and G. S. Watson have studied estimates of m of the form mₙ (x) = Yᵢk ( (x - Xᵢ) /aₙ) / k ( (x - Xᵢ) /aₙ). For distinct points x₁, , xₖ, we establish conditions under which (naₙ) ^1{2} (mₙ (x₁) - m (x₁), , mₙ (xₖ) - m (xₖ) ) is asymptotically multivariate normal.