Two Approaches to the Isotonic Change-Point Problem: Nonparametric and Minimax

Abstract

A change in model parameters over time often characterizes major events. Situations in which this may arise include observing increasing temperatures, intense rainfall, and the valuation of a stock. The question is whether these observations are simply the result of natural variation, or rather are indicative of an underlying monotonic trend. This is known as the isotonic change-point problem. Two approaches to this problem are considered: Firstly, for correlated data with short-range dependence, we prove that a particular U-statistic based on a modified version of the Jonckheere-Terpstra test statistic is asymptotically equivalent to a more complex U-statistic discussed by Shen and Xu (2013); one that has been shown to outperform other existing tests in a variety of situations. Secondly, we shall justify and utilize the minimax criterion in order to identify the optimal test statistic within a specified class. We shall see that, as motivated by the projection method, the aforementioned class is the class of contrasts. It shall be proven that the set of coefficients originally proposed by Abelson and Tukey (1963), and utilized by Brillinger (1989) in the isotonic change-point setting, are in fact minimax in the independent data case. For correlated data with shortrange dependence, we shall demonstrate a sufficient condition for minimaxity to hold.