Minimax Rates for Learning Smooth Populations of Parameters

2026-10-02

AI Summary

This study investigates the minimax convergence rates for pointwise estimation of smooth binomial mixture densities, with a focus on the trade-off between sample size and the number of trials. By exploiting the polynomial structure of the model, the authors derive theoretical lower bounds and identify critical barriers to estimation. Furthermore, under heterogeneous trial settings, they propose a regularized orthogonal series estimator based on the effective sample size to achieve matching upper bounds. The primary contribution of this work lies in establishing tight minimax upper and lower bounds that reveal a three-phase trade-off mechanism between sample size and the number of trials. Collectively, these results provide a complete characterization of the fundamental statistical limits governing the learning of smooth parametric populations.

Abstract

We study pointwise estimation of a smooth binomial mixing density and characterize its minimax rates. Assuming that the mixing density is $s$-H\"older smooth, we first consider the homogeneous setting with a common number of binomial trials $t$. We derive matching lower and upper bounds that reveal three regimes depending on the relative sizes of the sample size $n$ and the number of trials $t$. When $t$ is small, the finite number of trials creates an identification barrier that persists regardless of sample size; at intermediate $t$, more data reduce sampling uncertainty, while the number of trials limits how finely the mixing density can be recovered; and when $t$ is sufficiently large, the usual nonparametric density estimation rate is recovered. Our lower bounds exploit the polynomial structure of the binomial mixture model, while attainability is achieved by a suitably regularized orthogonal series estimator. We then extend the analysis to heterogeneous trial parameters, where the minimax rate depends on the full trial profile through the effective sample size available at each polynomial degree. These results characterize the fundamental statistical limits of learning smooth populations of binomial probabilities under both homogeneous and heterogeneous trials.

Problem

• minimax rates
• binomial mixing density
• pointwise estimation
• nonparametric density estimation
• smooth populations

Innovation

• minimax rates
• binomial mixture model
• orthogonal series estimator
• nonparametric density estimation
• heterogeneous trials