Two Sample Problem for Rounded Data

This paper deals with the two sample problem for rounded data in the i.i.d. model. It is well known that under the null hypothesis the two sample Kolmogorov-Smirnov statistic without rounding converges in distribution to the supremum of a standard Brownian bridge. We establish that a natural statistic of the Kolmogorov-Smirnov type based on the rounded data converges in distribution to the same limit as the full observation case. Our result is based on ``Donsker's theorem for discretized data'' given by Nishiyama (2008, J. Japan Statist. Soc.).