Consistent statistics for estimating and testing hypotheses from grouped samples

This paper deals with the problem of making statistical inferences about an unknown parameter in the case when the sample is grouped at the mid-points of equispaced intervals. The main discussion is concerned with a method which leads to consistent statistics, and with the asymptotic efficiency of these statistics compared to (a) the same statistics with ungrouped data, and (b) the maximum likelihood statistics. Unlike previous corrections or grouping, due to Sheppard and Lindley, the analysis is valid for grouping intervals of arbitrarily large width, and numerical studies are given for several examples.