Abstract: Let (X_i, Y_i), i=1, …, n be R^d×R¹-valued iid. samples taken from (X, Y). We denote the regression function by m(x)=E(Y|X=x). In this paper we consider the Mean Square Error (MSE) of two usual estimates of m(x₀) based on (X_i, Y_i), i=1,…, n at a point x₀, the Uniform-Kernel Estimate m_n(x₀) and NN-Estimate m_n(x₀). We prove that under some reasonable conditions the lowest asymptotic MSE attained for these two estimates have the same form:
C(x₀)n^-4/(d+4)+o(n^-4/(d+4)), where C(x₀) is a constant depending on x₀. Hence from the point of view of MSE, one has no reason to claim superiority of m_n(x₀)to m_n(x₀) or vice versa.