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.

翻译：在实际应用中，四舍五入无处不在。虽然研究人员已探讨过连续随机变量取整的影响，但取整也可能应用于离散随机变量的函数。例如，为推断国家紧急状态后的超额自杀死亡人数，相关机构可能提供紧急状态前后死亡人数的四舍五入平均值。全球自杀率普遍较低，这种取整可能严重影响对自杀率变化的推断。本文研究当取整至最近整数的平均值作为非负离散随机变量代理变量的情形。具体而言，当我们将U = n[Y /n]作为Y的代理变量时，我们的目标是对Y的概率质量函数参数进行推断。U的概率生成函数、E(U)和Var(U)反映了Y支撑集粗化的效应。此外，本文还针对若干特殊情形探讨了矩与分布参数估计量。我们证明在特定条件下，取整影响微乎其微。但通过三个实例也发现，取整可能对统计推断产生显著影响。本文提出的简易方法能部分抵消取整误差效应。针对某些概率分布难以推导出关于U的极大似然估计量的问题，我们提供了数值求解估计量的框架。