Optimal Rates for Estimation of Two-Dimensional Totally Positive Distributions

TL;DR

This paper investigates the optimal estimation rates for two-dimensional totally positive distributions, showing that additional total positivity constraints lead to faster minimax rates and providing efficient algorithms validated by simulations.

Contribution

It introduces the first analysis of minimax estimation rates for 2D totally positive distributions with smooth densities, highlighting the benefits of the total positivity constraint.

Findings

Polynomially faster minimax rates with total positivity

Development of fast algorithms for estimator computation

Simulation studies confirming theoretical estimation rates

Abstract

We study minimax estimation of two-dimensional totally positive distributions. Such distributions pertain to pairs of strongly positively dependent random variables and appear frequently in statistics and probability. In particular, for distributions with $\beta$-H\"older smooth densities where $\beta \in (0, 2)$, we observe polynomially faster minimax rates of estimation when, additionally, the total positivity condition is imposed. Moreover, we demonstrate fast algorithms to compute the proposed estimators and corroborate the theoretical rates of estimation by simulation studies.