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.