A test for detecting outlying cells in the multinomial distribution and Two-way contingency tables

A test based on the maximum adjusted residual from multinomial models, namely, the M test, is proposed. Sharp bounds on its critical values are provided for the multinomial case, while for the two-way table we present conservative critical values. The new test can be used to test null hypotheses against one- or two-sided alternatives and to detect outliers simultaneously with the rejection of the null hypothesis. Under certain alternatives, the M test is asymptotically more powerful than the chi-squared test.