Smoothing ordered sparse contingency tables and the Chi-Squared test

To estimate cell probabilities for ordered sparse contingency tables several smooth- ing techniques have been investigated. It has been recognized that nonparametric smoothing methods provide estimators of cell probabilities that have better performance than the pure frequency estimators. With the help of simulation examples it is shown in this paper that these smoothing techniques may help to get test which are more powerful than Chi-Squared test with raw data. But the distribution of the Chi-Squared statistics after smoothing is unknown. This distribution can also be estimated by simulation methods.