Estimation and reconstruction for zero-one Markov processes

Given a Markov process with state space {0, 1} we treat parameter estimation of the transition intensities and state estimation of unobserved portions of the sample path, based on various partial observations of the process. Parameter estimators are devised and shown to be consistent and asymptotically normal. State estimators are computed explicitly and represented in recursive form. Observation mechanisms include regularly spaced samples, regular samples with time jitter, Poisson samples. Poisson samples with state 0 unobservable, observability defined by an alternating renewal process, averaged samples, observation of transition times into state 1 and observation of a random time change of the underlying process. The law of the observability process may be partly unknown. The combined problem of state estimation with estimated parameters is also examined.