Predicting Productivity of Hockey Players via Mixture Models (Empirical Bayes Methodology)

Abstract

The objective of this thesis is to show how advanced methods based on mixture models can be used to predict the productivity of hockey players, measured by the rate at which they produce goals and assists. The performance of the methods is evaluated on existing data from one full National Hockey League (NHL) season. Over a large time frame, such predictions come fairly easily, regardless of the method we choose. However, our focus is on predictions obtained from relatively -- sometimes even significantly -- short sampling periods. If we look solely at the first 3-5 weeks of the season, the naïve estimator, based on maximum likelihood, is essentially useless at predicting how someone will perform for the remainder of the year. Simple methods such as ``one-fits-all'' estimators and naïve shrinkage estimators represent an improvement, but it turns out we can do better. We look at both parametric and nonparametric empirical Bayes approaches to fitting mixture models, with the objective of showing that these methods provide good predictions in a small time frame. In particular, we will cover two competitive approaches, the Poisson-Gamma parametric model and the Kiefer-Wolfowitz nonparametric method. Both of them construct certain mixtures of Poisson distributions, but contrary to the setting prevailing in the literature, we have to deal with the fact that our Poisson outcomes are for different players observed over different time periods, depending on the number of games played, or the total amount of time spent on ice.