## Annals of Statistics

- Ann. Statist.
- Volume 48, Number 2 (2020), 1072-1097.

### Worst-case versus average-case design for estimation from partial pairwise comparisons

Ashwin Pananjady, Cheng Mao, Vidya Muthukumar, Martin J. Wainwright, and Thomas A. Courtade

#### Abstract

Pairwise comparison data arises in many domains, including tournament rankings, web search and preference elicitation. Given noisy comparisons of a fixed subset of pairs of items, we study the problem of estimating the underlying comparison probabilities under the assumption of strong stochastic transitivity (SST). We also consider the noisy sorting subclass of the SST model. We show that when the assignment of items to the topology is arbitrary, these permutation-based models, unlike their parametric counterparts, do not admit consistent estimation for most comparison topologies used in practice. We then demonstrate that consistent estimation is possible when the assignment of items to the topology is randomized, thus establishing a dichotomy between worst-case and average-case designs. We propose two computationally efficient estimators in the average-case setting and analyze their risk, showing that it depends on the comparison topology only through the degree sequence of the topology. We also provide explicit classes of graphs for which the rates achieved by these estimators are optimal. Our results are corroborated by simulations on multiple comparison topologies.

#### Article information

**Source**

Ann. Statist., Volume 48, Number 2 (2020), 1072-1097.

**Dates**

Received: October 2017

Revised: October 2018

First available in Project Euclid: 26 May 2020

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1590480046

**Digital Object Identifier**

doi:10.1214/19-AOS1838

**Mathematical Reviews number (MathSciNet)**

MR4102688

**Subjects**

Primary: 62J15: Paired and multiple comparisons 62F07: Ranking and selection

**Keywords**

Pairwise comparisons strong stochastic transitivity structured matrix completion

#### Citation

Pananjady, Ashwin; Mao, Cheng; Muthukumar, Vidya; Wainwright, Martin J.; Courtade, Thomas A. Worst-case versus average-case design for estimation from partial pairwise comparisons. Ann. Statist. 48 (2020), no. 2, 1072--1097. doi:10.1214/19-AOS1838. https://projecteuclid.org/euclid.aos/1590480046

#### Supplemental materials

- Supplement to “Worst-case versus average-case design for estimation from partial pairwise comparisons”. Due to space constraints, we have relegated the technical details of remaining proofs to the supplement [38]. The supplement also contains a section characterizing the minimax denoising error under partial observations.Digital Object Identifier: doi:10.1214/19-AOS1838SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.