  In practice, the use of rounding is ubiquitous. Although researchers have
looked at the implications of rounding continuous random variables, rounding
may be applied to functions of discrete random variables as well. For example,
to infer on suicide excess deaths after a national emergency, authorities may
provide a rounded average of deaths before and after the emergency started.
Suicide rates tend to be relatively low around the world and such rounding may
seriously affect inference on the change of suicide rate. In this paper, we
study the scenario when a rounded to nearest integer average is used as a proxy
for a non-negative discrete random variable. Specifically, our interest is in
drawing inference on a parameter from the pmf of Y , when we get U = n[Y /n] as
a proxy for Y . The probability generating function of U , E(U ), and Var(U )
capture the effect of the coarsening of the support of Y . Also, moments and
estimators of distribution parameters are explored for some special cases. We
show that under certain conditions, there is little impact from rounding.
However, we also find scenarios where rounding can significantly affect
statistical inference as demonstrated in three examples. The simple methods we
propose are able to partially counter rounding error effects. While for some
probability distributions it may be difficult to derive maximum likelihood
estimators as a function of U , we provide a framework to obtain an estimator
numerically.
