Akaike information criterion for segmented regression models

In segmented regression, when the regression function is continuous at the change-points that are the boundaries of the segments, it is also called joinpoint regression, and the analysis package developed by \cite{KimFFM00} has become a standard tool for analyzing trends in longitudinal data in the field of epidemiology. In addition, it is sometimes natural to expect the regression function to be discontinuous at the change-points, and in the field of epidemiology, this model is used in \cite{JiaZS22}, which is considered important due to the analysis of COVID-19 data. On the other hand, model selection is also indispensable in segmented regression, including the estimation of the number of change-points; however, it can be said that only BIC-type information criteria have been developed. In this paper, we derive an information criterion based on the original definition of AIC, aiming to minimize the divergence between the true structure and the estimated structure. Then, using the statistical asymptotic theory specific to the segmented regression, we confirm that the penalty for the change-point parameter is 6 in the discontinuous case. On the other hand, in the continuous case, we show that the penalty for the change-point parameter remains 2 despite the rapid change in the derivative coefficients. Through numerical experiments, we observe that our AIC tends to reduce the divergence compared to BIC. In addition, through analyzing the same real data as in \cite{JiaZS22}, we find that the selection between continuous and discontinuous using our AIC yields new insights and that our AIC and BIC may yield different results.