The Bradley–Terry model for paired comparisons is a simple and muchstudied means to describe the probabilities of the possible outcomes when individuals are judged against one another in pairs. Among the many studies of the model in the past 75 years, numerous authors have generalized it in several directions, sometimes providing iterative algorithms for obtaining maximum likelihood estimates for the generalizations. Building on a theory of algorithms known by the initials MM, for minorization–maximization, this paper presents a powerful technique for producing iterative maximum likelihood estimation algorithms for a wide class of generalizations of the Bradley–Terry model. While algorithms for problems of this type have tended to be custom-built in the literature, the techniques in this paper enable their mass production. Simple conditions are stated that guarantee that each algorithm described will produce a sequence that converges to the unique maximum likelihood estimator. Several of the algorithms and convergence results herein are new. 1. Introduction. In

A common statistical model for paired comparisons is the Bradley-Terry model. This research re-parameterizes the Bradley-Terry model as a single-layer artificial neural network (ANN) and shows how it can be fitted using the delta rule. The ANN model is appealing because it makes using and extending the Bradley-Terry model accessible to a broader community. It also leads to natural incremental and iterative updating methods. Several extensions are presented that allow the ANN model to learn to predict the outcome of complex, uneven two-team group competitions by rating individual players—no other published model currently does this. An incremental-learning Bradley-Terry ANN yields a probability estimate within less than 5 % of the actual value training over 3,379 multiplayer online matches of a popular teamand objective-based first-person shooter. Keywords: Bradley-Terry model, paired comparisons, neural networks, delta rule, probability estimates

We examine the problem of measuring influence based on the information contained in the data on communication between scholarly publications, judicial decisions, patents, webpages, and other entities. The measurement of influence is useful to address several empirical questions such as reputation, prestige, aspects of the diffusion of knowledge, the markets for scientists and scientific publications, the dynamics of innovation, ranking algorithms of search engines in the World Wide Web, and others. In this paper we ask why any given methodology is reasonable and informative applying the axiomatic method. We find that a unique ranking method can be characterized by means of four axioms: anonymity, invariance to citation intensity, homogeneity for two-journal problems, and consistency. This method is easily implementable and turns out to be different from those regularly used in social and natural sciences, arts and humanities, and computer science. We thank Yiu T. Poon and participants in Iowa State’s NFL seminar for very helpful comments. We also thank Debora Lewi and Salwa Hammami for their research assistance.

La revue canadienne de statistique

and the art of scienti®c publishing

Abstract A common statistical model for paired comparisons is the Bradley–Terry model. This research re-parameterizes the Bradley–Terry model as a singlelayer artificial neural network (ANN) and shows how it can be fitted using the delta rule. The ANN model is appealing because it makes using and extending the Bradley–Terry model accessible to a broader community. It also leads to natural incremental and iterative updating methods. Several extensions are presented that allow the ANN model to learn to predict the outcome of complex, uneven two-team group competitions by rating individuals—no other published model currently does this. An incremental-learning Bradley– Terry ANN yields a probability estimate within less than 5 % of the actual value training over 3,379 multiplayer online matches of a popular team- and objective-based first-person shooter. Keywords Bradley–Terry model Paired comparisons Neural networks Delta rule Probability estimates 1

ABSTRACT We attempt to identify the 25 most-cited statistical papers, providing some brief commentary on each paper on our list. This list consists, to a great extent, of papers that are on non-parametric methods, have applications in the life sciences, or deal with the multiple comparisons problem. We also list the most-cited papers published in 1993 or later. In contrast to the overall most-cited papers, these are predominately papers on Bayesian methods and wavelets. We briefly discuss some of the issues involved in the use of citation counts. KEY WORDS: Citations, history of statistics