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

Abstract

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 $m_n(x) = \sum Y_ik((x - X_i)/a_n)/\sum k((x - X_i)/a_n)$. For distinct points $x_1, \cdots, x_k$, we establish conditions under which $(na_n)^{\frac{1}{2}}(m_n(x_1) - m(x_1), \cdots, m_n(x_k) - m(x_k))$ is asymptotically multivariate normal.