Abstract
Use of the cumulative and doubly cumulative sum statistics are discussed as a unifying approach to the shape and change-point hypotheses. The shape hypotheses are useful for a dose response analysis where it is usually difficult to assume a rigid parametric model. They are closely related to the change-point hypotheses which are useful for detecting a change of a time series. The proposed statistics are the maximal contrast type and the cumulative chisquared type. As compared with the well known restricted maximum likelihood approach called the isotonic regression the proposed method is computationally simple and easy to extend to various problems including the non-normal distributions and the two-way data. In particular there are several variations of the two-way data with or without natural ordering in the row and/or columns. Several interesting procedures are introduced according to those variations. The doubly cumulative sum statistics are newly introduced for the analysis under the convexity assumption and an elegant calculation of probability is developed by the second order Markov property of the basic variables. Finally an application to the three-way data is mentioned.
