Estimating counts through an average rounded to the nearest non‐negative integer and its theoretical & practical effects

In practice, the use of rounding is ubiquitous. Although researchers have looked at the implications of rounding continuous random variables, rounding may also be applied to functions of discrete random variables. For example, to infer the number of excess deaths due to falls after a national emergency, authorities may only provide a rounded average of deaths before and after the emergency started. Deaths from falling tend to be relatively low in most places, and such rounding may seriously affect inference on the change in the rate of deaths. In this paper, we study drawing inference on a parameter from the probability mass function of a non‐negative discrete random variable Y$$ Y $$, when for rounding coarsening width h$$ h $$ we get U=h[Y/h]$$ U=h\left[Y/h\right] $$ as a proxy for Y$$ Y $$. We show that the probability generating function of U$$ U $$, E(U)$$ \mathrm{E}(U) $$, and Var(U)$$ \mathrm{Var}\kern.15em (U) $$ capture the effect of the coarsening of the support of Y$$ Y $$. Theoretical properties are explored further under some probability distributions. Moreover, we introduce two relative risks of rounding metrics to aid the numerical assessment of how sensitive the results may be to rounding. Under certain conditions, rounding has little impact. However, we also find scenarios where rounding can significantly affect statistical inference. The methods are applied to infer the probability of success of a binomial distribution and estimate the excess deaths due to Hurricane Maria. The simple methods we propose can partially counter rounding error effects.