Changepoint Analysis as a Method for Isotonic Inference

Abstract

Concavity and sigmoidicity hypotheses are developed as a natural extension of the simple ordered hypothesis in normal means. Those hypotheses give reasonable shape constraints for obtaining a smooth response curve in the non-parametric input±output analysis. The slope change and in¯ection point models are introduced correspondingly as the corners of the polyhedral cones de®ned by those isotonic hypotheses. Then a maximal contrast type test is derived systematically as the likelihood ratio test for each of those changepoint hypotheses. The test is also justi®ed for the original isotonic hypothesis by a complete class lemma. The component variables of the resulting test statistic have second or third order Markov property which, together with an appropriate nonlinear transformation, leads to an exact and very ef®cient algorithm for the probability calculation. Some considerations on the power of the test are given showing this to be a very promising way of approaching to the isotonic inference.

An algorithm for a new method of change-point analysis in the independent Poisson sequence

“…It should be noted that the model is a typical example of the monotone hypothesis satisfying 1 . On the contrary, every monotone contrast = ( 1 , ⋯ , ) ′ , ′ = 0, 1 ≤ ⋯ ≤ , = (1, ⋯ ,1) ′ can be expressed by a unique and positive linear combination of step change-point contrasts ( − , ⋯ , − , , ⋯ , ) with the first elements − and the last − elements , = 1, … , − 1,thus suggesting a close relationship between the monotone hypothesis and the step change-point model (Hirotsu and Marumo, 2002). The null hypotheses 0 (1) and ∆ (5) are of course equivalent.…”

“…Therefore, we consider here a general case of unequal spacing of events and denote the time or location of the th event by . Then, the convexity hypothesis was introduced in Hirotsu and Marumo (2002) as 1 : * ′ ≥ 0, with at least one inequality strong,…”

“…Then the maximal standardized accumulated statistic, max acc. 1, has been shown to be an appropriate statistic to detect both an increasing tendency and the change-point simultaneously, see Hirotsu and Marumo (2002) and Hirotsu (2013). It is based on the accumulated efficient score and belongs to the complete class of tests for the monotone alternative in the means of a Poisson sequence (Hirotsu, 1982).…”

Isotonic Inference

“…Interestingly, (3) defines a − 1 edges of the convex cone defined by the simple ordered alternative (1); see [24]. For other extensions, including tree-structured, star-shaped, unimodality, and symmetry models, the reader is referred to [3] and [43].…”

Multiplicity adjustment for temporal and spatial scan statistics using Markov property

“…For example, Worsley (1986) demonstrated that in a change point problem, the computational time for the distribution of the maximum could be reduced using the Markov property among statistics (see also Kuriki et al 2002, Hirotsu andMarumo 2002, andreferences therein). In this paper, we develop a similar computation technique by taking advantage of the Markov structure among scan statistics.…”