The identity of the zero-truncated, one-inflated likelihood and the zero-one-truncated likelihood for general count densities with an application to drink-driving in Britain

Abstract

For zero-truncated count data, as they typically arise in capture-recapture modelling, we consider modelling under one-inflation. This is motivated by police data on drink-driving in Britain which shows high one-inflation. The data, which are used here, are from the years 2011 to 2015 and are based on DR10 endorsements. We show that inference for an arbitrary count density with one-inflation can be equivalently based upon the associated zero-one truncated count density. This simplifies inference considerably including maximum likelihood estimation and likelihood ratio testing. For the drink-driving application, we use the geometric distribution which shows a good fit. We estimate the total drink-driving as about $2{,}300{,}000$ drink-drivers in the observational period. As $227{,}578$ were observed, this means that only about 10% of the drink-driving population is observed with a bootstrap confidence interval of 9%–12%.