Translator Disclaimer
October 2020 Asymptotic theory of sparse Bradley–Terry model
Ruijian Han, Rougang Ye, Chunxi Tan, Kani Chen
Ann. Appl. Probab. 30(5): 2491-2515 (October 2020). DOI: 10.1214/20-AAP1564

Abstract

The Bradley–Terry model is a fundamental model in the analysis of network data involving paired comparison. Assuming every pair of subjects in the network have an equal number of comparisons, Simons and Yao (Ann. Statist. 27 (1999) 1041–1060) established an asymptotic theory for statistical estimation in the Bradley–Terry model. In practice, when the size of the network becomes large, the paired comparisons are generally sparse. The sparsity can be characterized by the probability $p_{n}$ that a pair of subjects have at least one comparison, which tends to zero as the size of the network $n$ goes to infinity. In this paper, the asymptotic properties of the maximum likelihood estimate of the Bradley–Terry model are shown under minimal conditions of the sparsity. Specifically, the uniform consistency is proved when $p_{n}$ is as small as the order of $(\log n)^{3}/n$, which is near the theoretical lower bound $\log n/n$ by the theory of the Erdos–Rényi graph. Asymptotic normality and inference are also provided. Evidence in support of the theory is presented in simulation results, along with an application to the analysis of the ATP data.

Citation

Download Citation

Ruijian Han. Rougang Ye. Chunxi Tan. Kani Chen. "Asymptotic theory of sparse Bradley–Terry model." Ann. Appl. Probab. 30 (5) 2491 - 2515, October 2020. https://doi.org/10.1214/20-AAP1564

Information

Received: 1 June 2019; Revised: 1 October 2019; Published: October 2020
First available in Project Euclid: 15 September 2020

MathSciNet: MR4149535
Digital Object Identifier: 10.1214/20-AAP1564

Subjects:
Primary: 60F05
Secondary: 62E20, 62F12, 62J15

Rights: Copyright © 2020 Institute of Mathematical Statistics

JOURNAL ARTICLE
25 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.30 • No. 5 • October 2020
Back to Top