MTD models for aggregate data from higher order Markov chains

We consider two higher order models for aggregate data on a finite state space. In the first model, aggregate data are obtained from N i.i.d. individuals who follow Mixture Transition Distribution (MTD) Markov model of lag l. In the second model, aggregate data are modeled as a MTD Markov model based on multinomial thinning. In both the cases, it is shown that Conditional Least Square Estimators are CAN for a fixed N.