Nonparametric inference under shape constraints: past, present and future

TL;DR

The paper surveys nonparametric inference under shape constraints, presenting a projection-based framework that unifies the Grenander estimator for monotone densities and multivariate log-concave density estimation. It develops the theory and computation of log-concave projections, affine equivariance properties, Hölder-continuity-based risk bounds, and minimax rates, highlighting adaptation to smoothness and piecewise-log-affine structures. The discussion then extends these ideas to modern semiparametric tasks, notably linear regression via optimal convex M-estimation and applications in isotonic subgroup selection and conditional independence testing, with rigorous guarantees. The work outlines key open problems, including online updating, missing data handling, and boundary behavior, signaling rich future directions for shape-constrained inference in high-dimensional settings.

Abstract

We survey the field of nonparametric inference under shape constraints, providing a historical overview and a perspective on its current state. An outlook and some open problems offer thoughts on future directions.

Figures (6)

• Figure 2.1: Left: The empirical distribution $\mathbb{G}_n$ (red) of a sample of size $n=30$ from the $\mathrm{Exp}(1)$ distribution, whose distribution function is the blue curve. The solid black line is the least concave majorant $\mathbb{G}_n^*$ of $\mathbb{G}_n$. Right: The $\mathrm{Exp}(1)$ density (blue) and the Grenander estimator (black).
• Figure 3.1: Illustration of the log-concave projection $\psi^*(P)$, which is well-defined when $P \in \mathcal{P}_d$, despite the non-convexity of $\mathcal{F}_d$.
• Figure 3.2: A schematic picture of a tent function in the case $d=2$.
• Figure 3.3: The log-concave maximum likelihood estimator (left) and its logarithm (right) based on 1000 observations from a standard bivariate normal distribution.
• Figure 4.1: Left: The density quantile function $J_0$ and its least concave majorant $\hat{J}_0$ for a standard Cauchy density. Right: The corresponding score functions $\psi_0$ and $\psi_0^*$.

Theorems & Definitions (29)

• Proposition 2.1: samworth2025modern
• Theorem 2.2: samworth2025modern
• Lemma 2.3: marshall1970discussion
• Proof 1
• Corollary 2.4
• Theorem 2.5: birge1987estimatingbirge1989grenander
• Lemma 3.1: ibragimov1956composition
• Lemma 3.2
• Theorem 3.3: prekopa1973logarithmicprekopa1980logarithmic
• Corollary 3.4