Various climatic processes are thought to evolve as rapid, shift-like events, which points at the presence of nonlinear dynamics. Time series analysis of nonlinear processes, however, is not trivial, for example, because of the difficulty in coming up with a realistic random process as a viable null hypothesis. In this methodology paper we construct a basic two-parameter process of shift-like excursions in an excitable system with a threshold. We demonstrate that this stochastic process, in comparison with a specific one-parameter process, can better reproduce main features of the waiting time histogram of abrupt glacial climate events, the Dansgaard-Oeschger events, as seen in two paleoclimatic proxy records, the North Greenland Ice core Project (NGRIP) ice core and the Sofular stalagmite δ18O records. We use the two-parameter process to test some arguments that were proposed in the ongoing discussion of a possible solar role in triggering Dansgaard-Oeschger events. Using our approach, we suggest for future studies to generate time series of random events which can serve as a more plausible null hypothesis for Monte Carlo based statistical tests on the regularity of shift-like processes such as Dansgaard-Oeschger events.