On Local Moments

We introduce the concept of local moments for a distribution in p, p[greater-or-equal, slanted]1, at a point z[set membership, variant]p. Local moments are defined as normalized limits of the ordinary moments of a truncated version of the distribution, ignoring the probability mass falling outside a window centered at z. The limit is obtained as the size of the window converges to 0. Corresponding local sample moments are obtained via properly normalized ordinary sample moments calculated from those data falling into a small window. The most prominent local sample moments are the local sample mean which is simply the standardized mean vector of the data falling into the window, and the local covariance, which is a standardized version of the covariance matrix of the data in the window. We establish consistency with rates of convergence and asymptotic distributions for local sample moments as estimates of the local moments. First and second order local moments are of particular interest and some applications are outlined. These include locally based iterative estimation of modes and contours and the estimation of the strength of local association.