Inference on a Zero-inflated Bivariate Binomial Distribution Applicable to Baseball Data

It is common to encounter situations where two success probabilities are parameters of interest based on nested (two-stage) binary data that commonly occur in sports data. Under these circumstances two correlated and nested binomial random variables can be utilized for analysis. Analysis of discrete count data with excessive zeros has been developed using different zero-inflated statistical models that allow for frequent zero-valued observations. The zero-inflated binomial distribution is one of the models that can be adequate when the underlying data generation of non-zero values is based on a sequence of independent Bernoulli trials. In this article, we propose a zero-inflated bivariate binomial distribution that can be applied to nested bivariate data when both components are zero-inflated. Some theoretical properties of the model are investigated and default Bayesian procedures regarding prior elicitation are also addressed. Moreover, the Bayesian predictive distribution is derived based on a three-fold distribution to see how a future observation behaves. Extensive simulation studies are performed to support the theoretical results, and real datasets for Major League Baseball players are analyzed to illustrate the methodology developed in this paper.