Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Finite-Sample Valid Rank Confidence Sets for a Broad Class of Statistical and Machine Learning Models

Onrina Chandra, Min-ge Xie

TL;DR

This work develops a finite-sample, nonparametric framework for inferring ranks across K populations by extending the Repro-Samples approach to discrete rank parameters. It constructs level-$1-\alpha$ rank confidence sets via inversion of neighborhood constraints and introduces a data-adaptive, discordance-based candidate set to keep computation tractable while preserving coverage. The method is validated through case studies and extensive simulations across unknown distributions, regression-based rankings, and Plackett–Luce top-choice data, showing robust finite-sample coverage and sharper, interpretable rank intervals than bootstrap or asymptotic alternatives. The framework unifies multiple ranking settings and is demonstrated to be practically applicable to wealth quantiles, sports standings, joke rankings, and hospital performance with heteroscedastic data.

Abstract

Ranking populations such as institutions based on certain characteristics is often of interest, and these ranks are typically estimated using samples drawn from the populations. Due to sample randomness, it is important to quantify the uncertainty associated with the estimated ranks. This becomes crucial when latent characteristics are poorly separated and where many rank estimates may be incorrectly ordered. Understanding uncertainty can help quantify and mitigate these issues and provide a fuller picture. However, this task is especially challenging because the rank parameters are discrete and the central limit theorem does not apply to the rank estimates. In this article, we propose a Repro Samples Method to address this nontrivial inference problem by developing a confidence set for the true, unobserved population ranks. This method provides finite-sample coverage guarantees and is broadly applicable to ranking problems. The effectiveness of the method is illustrated and compared with several published large sample ranking approaches using simulation studies and real data examples involving samples both from traditional statistical models and modern data science algorithms.

Finite-Sample Valid Rank Confidence Sets for a Broad Class of Statistical and Machine Learning Models

TL;DR

This work develops a finite-sample, nonparametric framework for inferring ranks across K populations by extending the Repro-Samples approach to discrete rank parameters. It constructs level- rank confidence sets via inversion of neighborhood constraints and introduces a data-adaptive, discordance-based candidate set to keep computation tractable while preserving coverage. The method is validated through case studies and extensive simulations across unknown distributions, regression-based rankings, and Plackett–Luce top-choice data, showing robust finite-sample coverage and sharper, interpretable rank intervals than bootstrap or asymptotic alternatives. The framework unifies multiple ranking settings and is demonstrated to be practically applicable to wealth quantiles, sports standings, joke rankings, and hospital performance with heteroscedastic data.

Abstract

Ranking populations such as institutions based on certain characteristics is often of interest, and these ranks are typically estimated using samples drawn from the populations. Due to sample randomness, it is important to quantify the uncertainty associated with the estimated ranks. This becomes crucial when latent characteristics are poorly separated and where many rank estimates may be incorrectly ordered. Understanding uncertainty can help quantify and mitigate these issues and provide a fuller picture. However, this task is especially challenging because the rank parameters are discrete and the central limit theorem does not apply to the rank estimates. In this article, we propose a Repro Samples Method to address this nontrivial inference problem by developing a confidence set for the true, unobserved population ranks. This method provides finite-sample coverage guarantees and is broadly applicable to ranking problems. The effectiveness of the method is illustrated and compared with several published large sample ranking approaches using simulation studies and real data examples involving samples both from traditional statistical models and modern data science algorithms.

Paper Structure

This paper contains 30 sections, 9 theorems, 71 equations, 1 figure, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Challenges in Rank Inference
  3. Main goal of the paper
  4. Existing Approaches to Rank Inference
  5. Main Contributions
  6. Sample Data and Model Setup
  7. Notations
  8. Methodology Developments and Theories
  9. Level $1-\alpha$ Confidence Set for Rank Vectors
  10. Refined confidence set using a candidate set with high coverage
  11. High–probability inclusion in the candidate set:
  12. Expected size of the Sub-Gaussian candidate set and discussion
  13. Validating Proposed Method via Case Studies
  14. Ranking Quantiles of Completely Unknown Distributions
  15. Ranking in Competitive Sports via a Regression Model
  16. ...and 15 more sections

Key Result

Theorem 1

Let $\bm R|_{\mathcal{I}}^{(0)}$ be the true rank vector for the populations indexed by $\mathcal{I}$. If the model $\bm R^{(0)}=\mathcal{S}\bigl(H(\mathcal{D},\bm U)\bigr)$ holds and the Borel-set constraint Tuprob is exact, then More generally if, $\mathbb P_{\bm U}\bigl(T(\bm U,\bm\theta)\in B_\alpha(\bm\theta)\bigr) \ge (1-\alpha)\bigl(1+o(\delta')\bigr),$ then for $\delta'>0$ which may depen

Figures (1)

  • Figure 1: Confidence intervals and rank sets for A1c control across 78 hospitals. Each black point denotes the observed rank of the estimated log-odds.

Theorems & Definitions (17)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Corollary 1
  • Lemma 3
  • proof : Proof of Lemma 1
  • proof : Proof of Lemma 2
  • proof : Proof of Corollary 1
  • Lemma 4
  • proof : Proof of Lemma 4
  • ...and 7 more