# Local polynomial density ratio estimation

arXiv: https://arxiv.org/abs/2609.38412 · HTML: https://mlsanity.com/paper/2609.38412/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 46 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 12 of 12)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Likelihood-free inference by ratio estimation](https://mlsanity.com/paper/1611.10242/) (2016)
- [Efficient multivariate entropy estimation via k-nearest neighbour distances](https://mlsanity.com/paper/1606.00304/) (2016)
- [Direct Estimation of Information Divergence Using Nearest Neighbor Ratios](https://mlsanity.com/paper/1702.05222/) (2017)
- [Adaptive learning of density ratios in RKHS](https://mlsanity.com/paper/2307.16164/) (2023)
- [Estimating Unbounded Density Ratios: Applications in Error Control under Covariate Shift](https://mlsanity.com/paper/2504.01031/) (2025)
- [Simple Local Polynomial Density Estimators](https://mlsanity.com/paper/1811.11512/) (2018)
- [Nearest Neighbor Sampling for Covariate Shift Adaptation](https://mlsanity.com/paper/2312.09969/) (2023)
- [Efficient functional estimation and the super-oracle phenomenon](https://mlsanity.com/paper/1904.09347/) (2019)
- [Boundary Adaptive Local Polynomial Conditional Density Estimators](https://mlsanity.com/paper/2204.10359/) (2022)
- [A new adaptive local polynomial density estimation procedure on complicated domains](https://mlsanity.com/paper/2308.01156/) (2023)
- [Multivariate root-n-consistent smoothing parameter free matching estimators and estimators of inverse density weighted expectations](https://mlsanity.com/paper/2407.08494/) (2024)

## Abstract (arXiv)

We propose a novel local-polynomial estimator of the ratio r=f/g of two d-dimensional densities f and g, from which independent samples are available. The estimator is shown to achieve pointwise minimax optimal rates over H"older classes of arbitrary smoothness index without additional logarithmic factors, and with smoothness being only assumed of r but not of f nor g. Our analysis remains valid for points on the boundary of the support of the distribution associated to g under a mild geometric assumption on the (unknown) boundary. We also derive a concentration inequality for the estimator, which can be useful in applications to classification, and give a rate in the supremum norm, where the supremum is also taken over boundary support points. The smoothness class over which the rate is obtained is sufficiently large that the individual densities f and g cannot be consistently estimated uniformly over this class. Direct estimators of the partial derivatives of r together with rates of convergence are also provided. We also obtain asymptotic normality of the estimators under quite general assumptions and again including boundary points, with a consistent variance estimator allowing for data-driven studentization. Moreover, we show how our estimator can be used to estimate the Kullback-Leibler information, in a construction which additionally uses debiasing. We provide the parametric rate together with asymptotic normality under sufficient smoothness of r relative to the dimension d.