  The paper considers parameter estimation in count data models using penalized
likelihood methods. The motivating data consists of multiple independent count
variables with a moderate sample size per variable. The data were collected
during the assessment of oral reading fluency (ORF) in school-aged children. A
sample of fourth-grade students were given one of ten available passages to
read with these differing in length and difficulty. The observed number of
words read incorrectly (WRI) is used to measure ORF. Three models are
considered for WRI scores, namely the binomial, the zero-inflated binomial, and
the beta-binomial. We aim to efficiently estimate passage difficulty, a
quantity expressed as a function of the underlying model parameters. Two types
of penalty functions are considered for penalized likelihood with respective
goals of shrinking parameter estimates closer to zero or closer to one another.
A simulation study evaluates the efficacy of the shrinkage estimates using Mean
Square Error (MSE) as metric. Big reductions in MSE relative to unpenalized
maximum likelihood are observed. The paper concludes with an analysis of the
motivating ORF data.
