Testing lumpability for marginal discrete hidden Markov models

When some states of a Markov chain are aggregated (or lumped) and the new process, with lumped states, inherits the Markov property, the original chain is said to be lumpable. We discuss the notion of lumpability for discrete hidden Markov models (DHMMs) and we explain why, in general, testing this hypothesis leads to non-standard problems. Nevertheless, we present a case where lumpability in DHMMs is a regular problem of comparing nested models. Finally, some simulation results assessing the performance of the proposed test and an application to two real data sets are given.