Confidence distributions for change-points and regime shifts

Abstract

Suppose observations y 1 ,...,y n stem from a parametric model f ( y,θ ), with the parameter taking one value θ L for y 1 ,...,y τ and another value θ R for y τ +1 ,...,y n . This article provides and examines two different general strategies for not merely estimating the break point τ but also to complement such an estimate with full confidence distributions, both for the change-point τ and for associated measures of differences between the two levels of θ . The first idea worked with involves testing homogeneity for the two segments to the left and the right of a candidate change-point value at a fine-tuned level of significance. Carrying out such a scheme requires having a goodness-of-fit test for constancy of the θ parameter over a segment of indices, and we also develop classes of such tests. These also have some independent interest. The second general method uses the log-likelihood function, profiled over the other parameters, and we show how this may lead to confidence inference for τ . Our methods are illustrated for four real data stories, with these meeting different types of challenges.