Confidence distributions for change-points and regime shifts

Suppose observations y1,…,yn stem from a parametric model f(y,θ), with the parameter taking one value θL for y1,…,yτ and another value θR for yτ+1,…,yn. 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.