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

Abstract

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, $x\in\lbrack a, b\rbrack = \lbrack 0, 1\rbrack$ say, by letting $\hat{F}_n(x), x \epsilon\lbrack 0, 1\rbrack$ be the unique cubic spline of interpolation to the sample cumulative distribution function $F_n(x)$ at equi-spaced points $x = jh, j = 0, 1,\cdots, l + 1, (l + 1)h = 1$, which satisfies specified boundary conditions $\hat{F}_n'(0) = a, \hat{F}_n'(1) = b$. The density estimate $\hat{f}_n(x)$ is then $\hat{f}_n(x) = d/dx\hat{F}_n(x)$. 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 $f^{(m)}\in\mathscr{L}_p\lbrack 0, 1\rbrack$. Then, for $m = 1,2,3$ and certain values of $p$, it is shown that $E(f_n(x) - f(x))^2 = O(n^{-(2m - 2/p)/(2m+1-2/p)}).$ Bounds for the constant covered by the "$O$" are given. An extension to the $\mathscr{L}_p$ case of known convergence properties of the derivative of an interpolating spline is found, as part of the proofs.