Interpolating Spline Methods for Density Estimation I. Equi-Spaced Knots

Statistical properties of a variant of the histospline density estimate introduced by Boneva-Kendall-Stefanov are obtained. The estimate we study is formed for x in a finite interval, x a, b = 0, 1 say, by letting Fₙ (x), x 0, 1 be the unique cubic spline of interpolation to the sample cumulative distribution function Fₙ (x) at equi-spaced points x = jh, j = 0, 1, , l + 1, (l + 1) h = 1, which satisfies specified boundary conditions Fₙ' (0) = a, Fₙ' (1) = b. The density estimate fₙ (x) is then fₙ (x) = d/dxFₙ (x). It is shown how to estimate a and b. A formula for the optimum h is given. Suppose f has its support on 0, 1 and f^ (m) ₚ 0, 1. Then, for m = 1, 2, 3 and certain values of p, it is shown that E (fₙ (x) - f (x) ) ² = O (n^- (2m - 2/p) / (2m+1-2/p) ). Bounds for the constant covered by the "O" are given. An extension to the Lₚ case of known convergence properties of the derivative of an interpolating spline is found, as part of the proofs.