# Penalty Specialists Among Goalkeepers: A Nonparametric Bayesian Analysis of 44 Years of German Bundesliga

Abstract: Penalty saving abilities are of major importance for a goalkeeper in modern football. However, statistical investigations of the performance of individual goalkeepers in penalties, leading to a ranking or a clustering of the keepers, are rare in the scientific literature. In this paper we will perform such an analysis based on all penalties in the German Bundesliga from 1963 to 2007. A challenge when analyzing such a data set is the fact that the counts of penalties for the different goalkeepers are highly imbalanced, leading to the question on how to compare goalkeepers who were involved in a disparate number of penalties. We will approach this issue by using Bayesian hierarchical random effects models. These models shrink the individual goalkeepers estimates towards an overall estimate with the degree of shrinkage depending on the amount of information that is available for each goalkeeper. The underlying random effects distribution will be modelled nonparametrically based on the Dirichlet process. Proceeding this way relaxes the assumptions underlying parametric random effect models and additionally allows to find clusters among the goalkeepers.

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### Cites background or result from "Penalty Specialists Among Goalkeepe..."

...This view is consistent with recent literature in football that focuses on the role of positional skills in determining the performance of both players and teams (e.g. Bornkamp et al., 2009; Seaton and Campos, 2011)....

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...Nonetheless, using game statistics has the advantage of ensuring the data’s objectivity, but it might not entirely reflect each player’s performance in a particular game (Bornkamp et al., 2009)....

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...Empirical evidence suggests that different roles require different skills to be successful (e.g. Bornkamp et al., 2009; Seaton and Campos, 2011)....

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6 citations

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##### References

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### "Penalty Specialists Among Goalkeepe..." refers background or methods in this paper

...A flexible and convenient solution is to use the Dirichlet process, dating back to [5]....

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...which allows for an efficient exact implementation in many cases (see [5] for details)....

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2,013 citations

### "Penalty Specialists Among Goalkeepe..." refers background in this paper

...Another reason for the popularity of Dirichlet process priors is the constructive stick-breaking representation of the Dirichlet process given by [17]....

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2,011 citations

### "Penalty Specialists Among Goalkeepe..." refers background in this paper

...For a random sample of size n from a probability distribution realized by a Dirichlet process [1] has shown that the prior density of the number of distinct values (clusters/components) k in n realizations is...

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