Isotonic, Convex and Related Splines

Posted on Oct 10, 2025

Tags: Isotonic, Convex, Spline

This note is for Wright, I. W., & Wegman, E. J. (1980). Isotonic, Convex and Related Splines. The Annals of Statistics, 8(5), 1023–1035.

The paper considered the estimation of isotonic, convex or related functions by means of splines.

• it is shown that certain classes of isotone or convex functions can be represented as a positive cone embedded in a Hilbert space.
• using this representation, it gives an existence and characterization theorem for isotonic or convex splines
• two special cases are examined showing the existence of a globally monotone cubic smoothing spline and a globally convex quintic smoothing spline
• finally, it examines a regression problem and show that the isotonic-type of spline provides a strongly consistent solution

Introduction

while a spline fit satisfies the requisite smoothness properties, it may not be isotone as desired

the paper presents a combined approach

Partial orders and isotonic splines

deal only with abelian groups of real functions which can be added pointwise

The relation, $»$, on the group $G$, of functions is required to satisfy the following conditions: (i) $f » f$ (ii) $f » g, g »h$ implies $f » h$ (iii) $f » g$ implies $f + h » g +h$ for all $h\in G$ (iv) $f » 0, g » 0$ implies $f + g » 0$ (v) $f » g$ and $g » f$ implies $f = g$

• a group satisfying (i) to (v) is called a partially ordered group
• if condition (v) is dropped, $G$ is called a preordered group

$P = {g\in G: g » 0}$ is called the positive cone of the order $»$

Let $L_2$ represent the set of functions on $[0, 1]$ which are Lebesgue measurable and square integrable with the usual Banach space norm

Let $W_m, m\ge 1$ be the set of functions on $[0, 1]$ for which $f^{(j)}, j = 0,1\ldots, m-1$ are absoultely continuous and $f^{(m)}$ is in $L_2$

This is a Hilbert space with inner produce $<f, g> = \sum_{j=0}^m\int_0^1 f^{(j)}(t)g^{(j)}(t)dt$.

Let $C^k, k=1,2,\ldots,\infty$ be the set of all functions on $[0, 1]$ which are $k$-times continuously differentiable

Let $F$ be a continuous linear map of $W_m$ into $W_1$ which commutes with the differentiation operator, i.e.,

\[D(Ff) = F(Df)\qquad \text{for all } f\in W_m\subset W_{m-1}\]

define a partial order $»$ on $W_m$ by $f » 0$ iff $(Ff)(t)\ge 0$ for every $t\in [0, 1]$

Examples:

1. If $F$ is the identity map, the set, $P = {g\in W_m: g» 0}$ is just the set of positive function $s$ in $W_m$
2. If $F = D$, $P$ is just the set of monotone nondecreasing functions in $W_m$. Similarly, if $F = -D$, $P$ is the set of nonincreasing functions.
3. If $F = D^2$, $P$ is the set of convex functions in $W_m$ while $F = -D^2$ yields the set of concave functions
4. If $F$ is defined by \(\begin{align*} Ff(t) & = Df(t) & 0 \le t < M\\ & = -Df(t) & M < t \le 1\,, \end{align*}\) then $P$ is the set of unimodal functions with mode $M$
5. If $F$ is defined by \(\begin{align*} Ff(t) &= - D^2f(t) & t_1 < t < t_2\\ &= D^2f(t) & 0\le t < t_1 \text{ or } t_2 \le t < 1 \end{align*}\) then $P$ is the set of functions which are concave on $[t_1, t_2]$ and convex elsewhere.

A well-studied problem in approximation theory is to find a solution of the following optimization problem:

\[\begin{align*} & \text{Minimize} \int_0^1 (f^{(m)}(t))^2dt \text{ subject to}\\ (a) & f\in W_m\\ (b) & f(t_i) = y_i, i=1,2,\ldots,n \end{align*}\]

the solution is called an interpolating spline

in contrast the problem

\[\begin{align*} &\text{Minimize} \sum_{i=1}^n (y_i - f(t_i))^2 + \lambda \int_0^1 (f^{(m)}(t))^2dt\\ \text{with $\lambda > 0$ fixed and subject to $f\in W_m$} \end{align*}\]

also has a spline function solution called a smoothing spline

a smoothing spline intermediate the above two formulations solve the following problem:

\[\begin{align*} &\text{Minimize} \int_0^1(f^{(m)}(t))^2dt \text{ subject to } \alpha_i\le f(t_i) \le \beta_i, \qquad i = 1,2,\ldots, n \end{align*}\]

A restricted isotonic spline

The general isotonic spline

Statistical interpretation

\[Y(t) = f(t) + e(t)\]

suppose $e(t)$ in a finite interval, say $[-e_1, e_2]$ containing 0

then $Y(t_i) + e_1 = \beta_i$ and $Y(t_i) - e_2 = \alpha_i$ forms a 100% CI for $f(t_i)$

the fact thatthe support is bounded allows to give 100% error bounds which implies that $f(t_i)$ always falls in $(\alpha_i, \beta_i)$, which in turn, guarantees the existence of $s_n$

if the support is unbounded, as with normal errors, finite 100% error bounds are impossible