On piecewise linear density estimators

We study piecewise linear density estimators from the L ₁ point of view: the frequency polygons investigated by S cott (1985) and J ones et al. (1997), and a new piecewise linear histogram. In contrast to the earlier proposals, a unique multivariate generalization of the new piecewise linear histogram is available. All these estimators are shown to be universally L ₁ strongly consistent. We derive large deviation inequalities. For twice differentiable densities with compact support their expected L ₁ error is shown to have the same rate of convergence as have kernel density estimators. Some simulated examples are presented.