  Lead time data is compositional data found frequently in the hospitality
industry. Hospitality businesses earn fees each day, however these fees cannot
be recognized until later. For business purposes, it is important to understand
and forecast the distribution of future fees for the allocation of resources,
for business planning, and for staffing. Motivated by 5 years of daily fees
data, we propose a new class of Bayesian time series models, a Bayesian
Dirichlet Auto-Regressive Moving Average (B-DARMA) model for compositional time
series, modeling the proportion of future fees that will be recognized in 11
consecutive 30 day windows and 1 last consecutive 35 day window. Each day's
compositional datum is modeled as Dirichlet distributed given the mean and a
scale parameter. The mean is modeled with a Vector Autoregressive Moving
Average process after transforming with an additive log ratio link function and
depends on previous compositional data, previous compositional parameters and
daily covariates. The B-DARMA model offers solutions to data analyses of large
compositional vectors and short or long time series, offers efficiency gains
through choice of priors, provides interpretable parameters for inference, and
makes reasonable forecasts.
