Review Swimming performance index based on extreme value theory Daniel T Gomes 1 and Lı ´gia Henriques-Rodrigues 2 Abstract The International Swimming Federation has developed a points system that allows comparisons of results between different events. Such system is important for several reasons, since it is used as a criterion to rank swimmers in awards and selection procedures of national teams. The points system is based entirely on the world record of the corres- pondent event. Since it is based on only one observation, this work aims to suggest a new system, based on the probability distribution of the best performances in each event. Using extreme value theory, such distribution, under certain conditions, converges to a generalized Pareto distribution. The new performance index, based on the peaks over threshold methodology, is obtained based on the exceedance probabilities correspondent to the swimmers’ times that exceed a given threshold. We work with 17 officially recognized events in 50m pool, for each women and men, and considered all-time rankings for all events until 31 December 2016. A study on the adequacy of the proposed generalized Pareto distribution index and a comparison between the performances of Usain Bolt and Michael Phelps are also conducted. Keywords Extreme quantiles, extreme value theory, generalized Pareto distribution, sports statistics, swimming Introduction Competitive swimming has a non-subjective ranking system, based on times. When one analyzes the results of a given event, such criterion is free from arbitrariness and is the universal way to choose the ‘‘best (fastest) swimmers.’’ However, for some reasons, it might be reasonable to compare performances in different events. For example, the winning time of the 100 m freestyle in a given competition is better than the win- ning time of the 200 m butterfly? Obviously, the concept of ‘‘better’’ needs to be defined. Intuitively, a great performance is the one that, in its respective event, corresponds to a discrepant result in comparison to the others. Such comparison is made by the International Swimming Federation (FINA) through a points system. It is important, since it is used to determine the winners of the World Cup. 1 It is also used by several national federations as the criterion to determine national teams, such as Brazil. 2 In 2015, the FINA points system was used for the first time to select the best performances of that year. In 2016, Katie Ledecky (United States) and Adam Peaty (Great Britain) were chosen the best performers of the year, by their performances in women’s 800 m freestyle and men’s 100 m breaststroke, respectively, in Rio de Janeiro Olympic Games. 3 The FINA points system is based on what is called the base times. 4 The base times are defined every year, based on the latest world record that was approved by FINA. The base times are defined with the cut date of 31 December. So, in a given event, let T i ðtÞ be the time obtained by the swimmer i, in seconds, at the year t, and Bðt  1Þ the base time, which is the world record, in Reviewer: John Einmahl (Tilburg University, Netherlands) Ian G. McHale (University of Liverpool, UK) Jonathan Tawn (Lancaster University, UK) 1 National School of Statistics, Brazilian Geography and Statistics Institute, Rio de Janeiro, Brazil 2 Department of Statistics, Institute of Mathematics and Statistics, University of Sa ˜o Paulo, Sa ˜o Paulo, Brazil Corresponding author: Daniel T Gomes, National School of Statistics, Brazilian Geography and Statistics Institute, Rio de Janeiro, Brazil. Email: daniel.gomes@ibge.gov.br International Journal of Sports Science & Coaching 0(0) 1–12 ! The Author(s) 2018 Article reuse guidelines: sagepub.com/journals-permissions DOI: 10.1177/1747954118808068 journals.sagepub.com/home/spo