A New and Flexible Approach to the Analysis of Paired Comparison Data

作     者：Oliveira, Ivo F. D. Ailon, Nir Davidov, Ori 

作者机构：UFVJM Fed Univ Valleys Jequitinhonha & Mucuri Dept Sci Engn & Technol Teofilo Otoni MG Brazil Technion Israel Inst Technol Dept Comp Sci Haifa Israel Univ Haifa Dept Stat Haifa Israel 

出 版 物：《JOURNAL OF MACHINE LEARNING RESEARCH》 (机器学习研究杂志)

年 卷 期：2018年第19卷第1期

页      面：2458-2486页

核心收录：

学科分类：08[工学] 0811[工学-控制科学与工程] 0812[工学-计算机科学与技术（可授工学、理学学位）] 

基　　金：European Research Council under European Union's Horizon 2020 Program ERC Israeli Science Foundation [1256/13, 457/17] European Research Council (ERC) Funding Source: European Research Council (ERC) 

主　　题：linear stochastic transitivity statistical ranking semi-definite programming model selection sensitivity analysis chess 

摘      要：We consider the situation where I items are ranked by paired comparisons. It is usually assumed that the probability that item i is preferred over item j is p(ij) = F(mu(i) - mu(j)) where F is a symmetric distribution function, which we refer to as the comparison function, and mu(i) and mu(j) are the merits or scores of the compared items. This modelling framework, which is ubiquitous in the paired comparison literature, strongly depends on the assumption that the comparison function F is known. In practice, however, this assumption is often unrealistic and may result in poor fit and erroneous inferences. This limitation has motivated us to relax the assumption that F is fully known and simultaneously estimate the merits of the objects and the underlying comparison function. Our formulation yields a flexible semi-definite programming problem that we use as a refinement step for estimating the paired comparison probability matrix. We provide a detailed sensitivity analysis and, as a result, we establish the consistency of the resulting estimators and provide bounds on the estimation and approximation errors. Some statistical properties of the resulting estimators as well as model selection criteria are investigated. Finally, using a large data-set of computer chess matches, we estimate the comparison function and find that the model used by the International Chess Federation does not seem to apply to computer chess.