  "For how many days during the past 30 days was your mental health not good?"
The responses to this question measure self-reported mental health and can be
linked to important covariates in the National Health and Nutrition Examination
Survey (NHANES). However, these count variables present major distributional
challenges: the data are overdispersed, zero-inflated, bounded by 30, and
heaped in five- and seven-day increments. To meet these challenges, we design a
semiparametric estimation and inference framework for count data regression.
The data-generating process is defined by simultaneously transforming and
rounding (STAR) a latent Gaussian regression model. The transformation is
estimated nonparametrically and the rounding operator ensures the correct
support for the discrete and bounded data. Maximum likelihood estimators are
computed using an EM algorithm that is compatible with any continuous data
model estimable by least squares. STAR regression includes asymptotic
hypothesis testing and confidence intervals, variable selection via information
criteria, and customized diagnostics. Simulation studies validate the utility
of this framework. STAR is deployed to study the factors associated with
self-reported mental health and demonstrates substantial improvements in
goodness-of-fit compared to existing count data regression models.
