Statistical analysis of rounded data: measurement errors vs rounding errors (Q2313803)

Consider the measurements in which the discretization step is constant and without loss of generality is equal to 1. The measured value \(X\) is rounded to the closest integer \(X^*\) according to the rule \(X^*=[X+0.5]\) where \([x]\) is the integer part of \(x\). Measurements of the quantity \(\mu\) are considered as i.i.d. random samples \(X_1,\dots,X_n\) with common mathematical expectation \(\mu\), the accuracy of measurements is characterized by their common standard deviation \(\sigma\). The estimate of \(\mu\) is obtained as:\[\hat{\mu}_n=n^{-1}\sum_{i=1}^n X^*_i.\] \par In the article, the upper bound for the limiting error \(|\hat{\mu}_n-\mu|\) as \(n \to \infty\) is proved. For finite \(n\) Monte-Carlo simulations are presented. It turned out that the error \(|\hat{\mu}_n-\mu|\) is comparable to 1 if \(\sigma \ll 1\) and vanishes with growing \(\sigma\) if \(\sigma> 1\). This means that one does not need the accuracy of measurements higher than the accuracy of rounded data.