Bayesian Approach to the Estimation of a Poisson Mean with Application to NBA Three Point Attempts

Patricia Pepple Williamson

doi:10.3844/jmssp.2018.219.231

Abstract

Bayesian Credible Intervals are proposed for a Poisson mean. These intervals are compared to five classical confidence intervals found in the literature. A simulation study is performed to compare the procedures using two different criteria and it is attempted to determine which of the procedures performs best for various values of the parameter and sample size. Estimation of the number of three-point shot attempts and three-point shot makes by the San Antonio Spurs is given as an example.

Highlights

During the 2011-2012, 2012-2013 and 2013-2014 regular seasons, the San Antonio Spurs were in the top three in team three point shooting percentage
Because a random phenomenon for which a count of some type may be modeled by a Poisson distribution, the number of three-point attempts and the number of three pointers made in a game are possible candidates
Because it seems reasonable to assume that these two counts are such that at most one three point attempt can occur in a very small interval of time and the number of attempts in two equal nonoverlapping time intervals should be independent and have the same distribution, the Poisson distribution seems like an appropriate model for each of these counts

Summary

Introduction

During the 2011-2012, 2012-2013 and 2013-2014 regular seasons, the San Antonio Spurs were in the top three in team three point shooting percentage. Khamkong (2012) compares four confidence intervals using coverage rate and estimated expected length; the scores interval is considered along with a proposed adapted Wald interval which outperformed the other intervals for small mean and small to moderate sample sizes. Letting tα/2,n-1 denote the 100(1α/2)th percentile of a t(n-1) distribution and noting that P[-tα/2,n-1 < Q2 < tα/2,n-1] = 1-α, the resulting 100(1α)% confidence interval for θ is given by: X

Results

Conclusion