Histosplines with Knots Which are Order Statistics

Summary The histospline considered here is a quadratic spline density estimate analogous to the Boneva–Kendall–Stefanov (BKS) histospline, with their equally spaced knots replaced by knots at every knth-order statistic. This estimate can be expected to be relatively more “flexible” where the density is greatest. We obtain “mean square error at a point” convergence rates. The rates obtained are uniform over classes of densities which have first (alternatively second or third) derivatives in a bounded set in ℒp. If kn is chosen optimally, then this estimate shares the same “near optimal” convergence rates of certain BKS estimates when the knot spacing is chosen optimally, kernel estimates when the scale factor is chosen optimally, and orthogonal series estimates when the length of the series is chosen optimally.