C. Hirotsu (University of Tokyo, Japan)
A relationship between the isotonic inference and the changepoint analysis
The isotonic inference has many applications in industrial problems where there is a natural ordering in the levels of a treatment such as dose, temperature, time and so on. A changepoint model is also essential in the industrial process control. In the present paper we first demonstrate a relationship between the monotone hypothesis and the step type changepoint model in the normal means. It is simply that each of the corner vectors of the convex cone defined by the monotone hypothesis corresponds to the component hypothesis of the changepoint model. On the other hand a complete class of tests for the monotone hypothesis is shown to be all the tests that are increasing in every element of the projections of the observation vector onto those corner vectors. Then it happens that a statistic called max t and defined by the standardized maximum of those projections has been developed independently in two different streams of the isotonic inference and the changepoint analysis. It is actually the likelihood ratio test (lrt) statistic for the changepoint hypothesis. Those considerations are extended to various isotonic hypotheses including convexity, sigmoidicity and two-way ordered alternatives which induce slope change, inflection and two-way changepoint models as their corner vectors, respectively. The lrt for those changepoint hypotheses are easily derived and they become appropriate tests also for the original isotonic hypotheses by virtue of the complete class lemma. An exact and very efficient algorithm is introduced for calculating the distribution function of the max t type statistics. This will thus give a systematic way of approach to the isotonic inference other than the isotonic regression which is often too complicated excepting for the monotone hypothesis. Some power comparisons will also be given.