A Unifying Approach to the Shape and Changepoint Hypotheses
Chihiro Hirotsu
Collaborative Research Center, Meisei University, Hino, Tokyo, Japan

Usually a changepoint model assumes a step-type change at some point of a time series. Then as an interesting thing it has been shown that the max acc. t test for the isotonic hypothesis is also appropriate for detecting the changepoint. It comes from a relationship that each corner vector of the polyhedral cone defined by the isotonic hypothesis corresponds to a component of the changepoint model. The max acc. t is essentially the maximal component of the projections of the observation vector on to the corner vectors of the polyhedral cone and a very efficient and exact algorithm is obtained based on the Markov property of the component statistics, see Hirotsu et al. (1992) and also Worsley (1986). The idea has been extended in the case of normal model to the concavity and the slope change hypotheses in Hirotsu and Marumo (2002).

In this paper we will extend the idea further to a Poisson model. We propose a maximal contrast type statistic for a concavity hypothesis based on the complete class lemma in Hirotsu (1982), which is also an efficient score test for a slope change model. A second order Markov property of the serial component statistics is shown and a factorization of the null distribution into the products of conditional probabilities is obtained. Based on it a very efficient and exact algorithm for the calculation of p-value is obtained as well as that of the moments. An interesting application will be in the monitoring of the time series of the spontaneous reporting of the drug-adverse event combination. For the purpose the max acc. t has been proved to have high power for detecting a step-type change and/or an increasing tendency of the frequencies of the reporting. However, after detecting a change and taking necessary actions it should be interesting to verify that the increasing tendency has been changed into decreasing, and the proposed method should be useful for the purpose. Another important application will be in the non-parametric input-output analysis. The method is also useful as a goodness of fit test of linearity. An extension to the exponential family is also mentioned.


Hirotsu, C. (1982). Use of cumulative efficient scores for testing ordered alternatives in discrete models. Biometrika 69, 567-577.

Hirotsu, C., et al. (1992). Multiple comparsion procedures based on the maximal component of the cumulative chi-squared statistic. Biometrika 79, 381-392.

Hirotsu, C. and Marumo, K. (2002). Changepoint analysis as a method for isotonic inference. Scand. J. Statist. 29, 125-138.

Worsley, K. J.(1986). Confidence regions and tests for a change-point in a sequence of exponential family of random variables. Biometrika 73, 91-104.

Keywords: Doubly accumulated statistic; Exponential family; Recursion formula; Second order Markov property

Biography: 1968 Doctor of Engineering, University of Tokyo

1972-1986 Associate Professor, University of Tokyo

1986-2000 Professor, University of Tokyo

2000 Emeritus Professor, University of Tokyo

2000-2010 Professor, Meisei University

2010 ∼ Collaborative Research Center, Meisei University