A unifying approach to the shape and change-point hypotheses in the discrete univariate exponential family

A unifying approach to the shape and change-point hypotheses is extended generally to a discrete univariate exponential family. The maximal contrast type tests are newly proposed for the convexity and sigmoidicity hypotheses based on the complete class lemma of tests for the restricted alternatives. Those tests are also efficient score tests for the slope change-point and inflection point models, respectively. For each of those tests the successive component statistics are the doubly- and triply-accumulated statistics. They have nice Markov properties for the exact and efficient recursion formulae for calculating the p-value. Further the sum of squares of the component statistics are developed as the cumulative chi-squared statistics for the directional goodness-of-fit tests of the dose-response model. Therefore the interesting applications will be in monitoring of spontaneous reporting of the adverse drug events and the directional goodness-of-fit tests.