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<p>The paper revisits popular models for football goal distribution modeling and proposes a new approach based on non-trivial finite mixture models (FMMs) that, to the best of our knowledge, have not yet been systematically applied in football modeling. To make sure that our results are valid, we assembled a novel large-scale dataset, significantly larger than those used in previous studies, and compared models using cross-validation, evaluating both in-sample Akaike Information Criterion (AIC) and out-of-sample negative log-loss (NLL). We analyzed both the marginal distribution of single-team goals and the joint distribution of the entire match result. We found that a two-component mixture of negative binomial distributions describes observed data better than other models in terms of both in-sample AIC and out-of-sample NLL. These findings may be extended to other discrete-scoring sports and align well with research on FMMs in other domains with heterogeneous and heavy-tailed distributions, such as insurance, cybersecurity, and environmental science.</p>

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Finite mixture models for modeling football goal distributions

  • 期刊论文

  • 发表日期：

    2026年6月17日

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  • Artur Karimov,
  • Aleksandr Koshkin,
  • Denis Butusov

摘要

The paper revisits popular models for football goal distribution modeling and proposes a new approach based on non-trivial finite mixture models (FMMs) that, to the best of our knowledge, have not yet been systematically applied in football modeling. To make sure that our results are valid, we assembled a novel large-scale dataset, significantly larger than those used in previous studies, and compared models using cross-validation, evaluating both in-sample Akaike Information Criterion (AIC) and out-of-sample negative log-loss (NLL). We analyzed both the marginal distribution of single-team goals and the joint distribution of the entire match result. We found that a two-component mixture of negative binomial distributions describes observed data better than other models in terms of both in-sample AIC and out-of-sample NLL. These findings may be extended to other discrete-scoring sports and align well with research on FMMs in other domains with heterogeneous and heavy-tailed distributions, such as insurance, cybersecurity, and environmental science.